Queues in Hospitals: Semi-Open Queueing Networks in the QED Regime (QED = Quality- and Efficiency-Driven)
תוריM בבתי חוליM;! רשׁתות סטוכסטיות חצי! - פתוחות! בתחוM המשׁ!”י! (מכווN שׁירות! - יעילות!)
Ph.D. Research Proposal
January 2008
Galit Yom-Tov ([email protected])
Adviser: Avishai Mandelbaum
Faculty of Industrial Engineering and Management Technion – Israel Institute of Technology Abstract The aim of this thesis is to study queueing in Health Care systems. As a first stage, we concentrate on analyzing a Medical Unit with s nurses and n beds, which are partly/fully occupied by patients. This is a service system in which the patients need medical care, given them by the nurses, on condition that there is a bed available in the Medical Unit for hospitalization. We model this by a semi-open queueing network with multiple statistically identical customers and servers. The questions we address are: How many servers (nurses) are required (staffing), and how many fixed resources (beds) are needed (allocation) in order to minimize costs while sustaining a certain service level? We answer this by developing QED regime policies that are asymptotically optimal at the limit, as the number of patients entering the system (λ), the number of beds (n) and the number of servers (s) grows to infinity together. These approximations form out accurate for parameters values that are realistic in a hospital setting. To date, we have developed this kind of heavy-traffic approximation for the following service- level-objectives: (i) the probability of blocking, (ii) the probability of waiting, and (iii) the expected waiting time. We have also developed fluid limits for a closely-related open network, in which the restriction on n was relaxed, and thus the blocking phenomenon cannot occur. We generalized our findings to any semi-open network which has the following structure: one arrival stream (M/M/1), one service node with multiple identical servers (M/M/S), and any arbitrary finite number of delay nodes (M/M/∞). Lastly, we are proposing several ways to continue this research in some interesting and useful directions.
1 Contents
Abstract 1
List of abbreviations and notation5
1 Introduction 7 1.1 The structure of modern hospitals...... 7 1.2 Background on system design...... 9 1.2.1 Managing bed capacity...... 9 1.2.2 Managing work-force capacity...... 10 1.3 The QED (Quality- and Efficiency-Driven) regime...... 12 1.3.1 Introduction to the Halfin-Whitt (QED) Regime...... 12 1.3.2 QED queues in hospitals...... 13 1.3.3 QED queues in our Internal-Ward application...... 15 1.4 Other related models...... 17 1.5 Other operating regimes...... 17 1.6 Research objectives...... 18
2 Extended Nurse-to-Patient model 20 2.1 The medical unit: Internal Ward (IW)...... 20 2.2 The model...... 21 2.3 Alternative models...... 24 2.3.1 Proposal 1...... 24 2.3.2 Proposal 2...... 24 2.3.3 Proposal 3...... 24 2.3.4 Proposal 4...... 26
3 System measures 28 3.1 Probability of blocking...... 28 3.2 Probability of waiting more than t units of time and the expected waiting time... 29 3.3 Probability of delay...... 31 3.4 Average occupancy level...... 31
4 The QED regime 32
5 Heavy traffic limits and asymptotic analysis in the QED regime 34 5.1 Approximation of the probability of delay...... 35 5.2 Approximation of the expected waiting time...... 38 5.3 Approximation of blocking probability...... 39
6 Comparison of approximations and exact calculations 40
7 Comparison with other models 44 7.1 The M/M/S/infinity/n system...... 44 7.2 Call center with IVR (Interactive Voice Response)...... 46
2 8 Generalizations 47 8.1 The marginal distribution...... 48 8.2 The probability of delay...... 48 8.3 The probability of blocking...... 49 8.4 Approximation of E[W]...... 50
9 Fluid limits 51 9.1 Steady-state analysis of the fluid system...... 54 9.2 Transient analysis of the fluid system...... 55
10 Defining optimal design 61
11 Further research 62 11.1 Near Future...... 62 11.2 Combining managerial / psychological / informational diseconomies-of-scale effects. 63 11.3 Phases of treatment or Heterogeneous patients...... 65 11.3.1 Combining the phases of treatment during hospitalization period...... 65 11.3.2 The influence of time delays before and after medical analysis or surgery... 67 11.3.3 Classes of patients (Heterogeneous patients)...... 67 11.4 Nurses in the QED regime, and doctors in the ED regime...... 68 11.5 The combination of patient-call treatments and nurse-initiated treatments...... 69 11.6 Integrating IWs with the EW as in Jennings and V´ericourt...... 70 11.7 Two service stations...... 70
A Appendix A 71
B Four auxiliary lemmas 72 B.1 Proof of Lemma1...... 72 B.2 Proof of Lemma2...... 73 B.3 Proof of Lemma3...... 76 B.4 Proof of Lemma4...... 79
C Proof of approximation of the expected waiting time 82
List of Figures
1 The basic operational model of a hospital system...... 8 2 Jennings and V´ericourt’s model [29, 28]...... 14 3 The MU model as a semi-open queueing network...... 15 4 The IW model as a semi-open queueing network...... 21 5 The IW model as a closed Jackson network...... 22 6 Alternative model - Proposal 1...... 25 7 Alternative model - Proposal 2...... 25 8 Alternative model - Proposal 3...... 26 9 Alternative model - Proposal 4...... 27 10 Three example of comparison between approximation and exact calculation - Small system...... 40 11 Two example of comparison between approximation and exact calculation - Medium system...... 41 12 Comparison of approximation and exact calculation - Large system...... 42
3 13 Comparison of approximation and exact calculation - Israely Hospital...... 43 14 Comparison of approximation and exact calculation - r = 0.25...... 43 15 Version 3...... 51 16 Fluid DE convergence...... 58 17 Phases of hospitalization - Model 1...... 66 18 Phases of hospitalization - Model 2...... 66 19 New model for two classes of patients...... 68 20 ED (doctors) and QED (nurses) model...... 69 21 New model for nurse staffing and bed allocation according to Jennings and V´ericourt (2007)...... 70
List of Tables
1 Parameters for small systems...... 40 2 Parameters for medium systems...... 41 3 Parameters for a large system...... 41 4 Parameters based on data from Israely Hospital...... 42 5 Parameters based on Jennings and V´ericourt’sarticle...... 43 6 Parameters for illustration of the fluid DE convergence...... 57
4 List of abbreviations and notation
Abbreviations
MU Medical Unit EW Emergency Ward IW Internal Ward IT Information Technology LoS Length of Stay NRP Nurse Rostering Problem NSP Nurse Scheduling Problem QED Quality- and Efficiency-Driven ED Efficiency Driven QD Quality Driven RN Registered Nurse i.i.d. independent and identically distributed FCFS First Come First Served RFID Radio Frequency IDentification IVR Interactive Voice Response DE Differential Equation FLLN Functional Law of Large Numbers u.o.c. uniformly on compact a.s. almost surely K diag{a}, a ∈ R The matrix diag {a1, ..., aK } ∂θ(q), θ : K → K [ ∂θj (q) ]K R R ∂qk j,k=1 HRM Human Resources Management QoS Quality of Service
Notation
N(t) The number of needy patients at time t D(t) The number of dormant patients at time t C(t) The number of beds in cleaning at time t s Number of nurses n Number of beds
5 λ Arrival rate µ Service rate δ Dormant/activation rate γ Cleaning rate p Probability of staying in the medical unit after service π(i, j, k) The stationary probability of having i needy patients, j dormant patient
and k beds in cleaning (sometimes denoted πn(i, j, k) or πn,s(i, j, k)) A π (x − ei) The probability that the system is in state x − ei at the arrival epoch of a customer to node i
Pl The probability that there are l beds occupied in the system
P (blocked) The probability of blocking of the medical unit (P (blocked) = Pn) W The steady state in-queue waiting time, for a hypothetical newly needy patient pn(s, t) The tail of the steady state distribution of W OC(n, s) Average occupancy level R The offered load λ ρ The offered load per server; ρ = (1−p)sµ λ RN The solution of the balance equations for the needy state; RN = (1−p)µ pλ RD The solution of the balance equations for the dormant state; RD = (1−p)δ λ RC The solution of the balance equations for the cleaning state; RC = γ
≈ an ≈ bn if an/bn → 1, as n → ∞ φ(·) The standard normal density function Φ(·) The standard normal distribution function N(0, 1) A standard normal random variable with distribution function Φ E Expectation P Probability measure
6 1 Introduction
1.1 The structure of modern hospitals
A Hospital or Medical Center is an institution for health care, which is able to provide long-term patient stays. One can distinguish two types of patients: inpatients and outpatients. Some patients in a hospital come only for a diagnosis and/or therapy and then leave (’outpatients’), while others are ’admitted’ and stay overnight or for several weeks or months (’inpatients’). Hospitals are usually distinguished from other types of medical facilities by their ability to admit and care for inpatients. Within hospitals, the two types of patient are usually treated in separate systems, and thus can be analyzed separately. We will concentrate on the inpatient system. In the modern age, hospitals are a combination of several medical units (MU) specializing in different areas of medicine such as internal medicine, surgery, plastic surgery, and childbirth. In ad- dition to these medical units, the hospital includes some service units such as laboratories, imaging facilities, and IT (Information Technology), that provide service to the medical units. Typically, inpatients arrive to the hospital, randomly, via an Emergency Ward (EW), which deals with imme- diate threats to health and has the capacity to dispatch emergency medical services. For operational purposes, therefore, the flow of patients in a hospital can be viewed as in Figure1: a patient enters the EW, is treated, and then dismissed after treatment or admitted to stay, the latter if the doctors decided to hospitalize the patient and there is an available bed at an appropriate MU, in which case the patient is transferred to that MU. At some point in time (i.e., when the patient is cured or transfered to other medical centers, or unfortunately dies) the patient leaves the hospital system. Focusing on the operational point of view, the hospital includes doctors, nurses and administra- tive staff. Each MU is managed autonomously, with its own medical staff. Each MU has a limited capacity which is a function of the physical space (static capacity) and the staffing levels (dynamic capacity). The physical space is usually measured by the number of beds allocated to that MU, and the staffing levels by the number of service providers: doctors, nurses, and general workers (sanitation staff, etc.). Naturally, capacity restrictions can lead to a situation of system blocking. Thus the EW and MUs can be blocked, and a situation where ambulances are turned away [16], or a patient is waiting in the EW for assignment is not a rare event. In large medical centers there are several MUs of the same type, denoted as a parallel setting. This division is due to a combination of location constraints and the inability to manage large wards efficiently. Nevertheless, blocking does also occur in such large medical centers. All of the above leads one to the conclusion that one can model the hospital as a complex
7 The medical Center:
Services MU1
MU2
Emergency Patient Arrivals Ward MU3 discharge
Blocked Patient MUn patients discharge
Figure 1: The basic operational model of a hospital system
stochasticJennings network, where(2006): each node represents some process. We can then examine the flow of patients at that network, as shown in the case-study of Bruin et al. [13]. If that flow is not smooth then patients are getting stuck at various points in the system, waiting for medical care, waiting in queues. The most well known queues for medical care are those for surgery, organ transplants, s n-s very expensive diagnostic tests such as C.T. and MRI, and specialists. For some of the above, one Needy can wait for months [10]. Less noticed, though muchexp( moreμ ) common, are queues for hospital beds, doctors, nurses, lab tests and medication. These queues cause delays during treatment, when the response time can be critical for patient safetyn and quality of care (see Sobelov et al. [48]). Health-Care queueing research has the capabilityDormant to deal with various aspects of the Health-Care system. For example: exp( λ )
1. Scheduling - as the optimal scheduling of surgery rooms, in order to minimize wait while con- sidering diverse patients’ needs and system constraints; or managing out-patient appointments [25].
2. Routing - as the routing of patients from the EW to the MU [50].
3. Staffing - as how many nurses to assign to a MU [29], first at the planing stage and then dynamically.
4. Design - as capacity planning [21] and what is the optimal bed allocation [20].
5. Costing - as the optimal sharing of surgical costs in the presence of queues [18].
8 Some of these issues were noticed and approached in the past, usually not as a Health-Care problem, but in a more general perspective. But many aspects have not been treated, and some of them are crucial for Health-Care systems, such as adaptivity of large-system-approximations to small systems, and combination of medical and psychological aspects. One of the main methods used is simulation (see for example [41]). The reasons for the popularity of simulation in Health-Care seem to be just like those in call centers: there is a widening gap between the complexity of the modern Health-Care system and the analytical models available to accommodate this complexity. Moreover, simulation techniques are relatively simple user-friendly tools [17]. Other researchers use various methods of mathematical programing for modeling and analysis of Health-Care systems (see Halls’ book [25] for various works on the subject). Few tried to deal with the stochastic characteristics of the system using queueing theory as used for outpatient analysis and in other fields such as call centers. Nevertheless, already this little work suggests that stochastic-based insights could significantly advance our understanding of inpatient Health-Care systems.
1.2 Background on system design
Due to the complexity of the Health-Care system, capacity management decisions are carried out hierarchically. We will now shortly describe the whole process: forecasting demand, setting bed capacity at the hospital level, setting the allocation of beds inside each individual hospital, setting staffing levels, shifts scheduling and rescheduling. It is common practice to distinguish between static- and dynamic-capacity decisions; usually these decisions are the charter of different manage- ment teams. While static-capacity is hard to change and planning is made for long-term periods, dynamic-capacity is flexible, namely it can be adapted to changes in circumstances within a short period of time. In the literature overview on capacity planning in health care, presented by Smith- Daniels et al. [47], the following classification is proposed: facility resources planning (for example: bed allocation) is separated from work-force resource planning (such as nurses and doctors staffing and scheduling). In our literature review we use the same classification.
1.2.1 Managing bed capacity
Long-term capacity planning is based on forecasting the demand for inpatient services. The forecast is based on mathematical models (such as time series) that predict the changes in inpatient demand over long periods (i.e. months and years). For example, one can use the forecasting models of Jones et al. [30], or Kao and Tung [31]. Based on this prediction policy makers can deduce the required bed capacity of the hospital.
9 As in other service systems, in most Health-Care systems, the arrival rate of patients entering the system varies over time. Over short periods of time, minute-by-minute for example, there is significant stochastic variability in the number of arriving patients. Over longer periods of time, the course of the day, the days of the week, the months of the year – there can also be predictable variability, such as the seasonal patterns that arriving patients follow. Example of patterns in admission of cardiac inpatients into EW, during an average day, can be found in de Bruin et al. [13]. Harper and Shahani [26] have shown another example of changes in mean bed occupancy through the months of the year of adult medical (as opposed to surgical) population, in a major UK NHS Trust; this pattern could reflect the pattern of admissions, assuming that the bed capacity was fixed during that period. Because the service capacity cannot be inventoried, one should vary the number of available beds and medical staff in the short-term, to track the predictable variations in the arrival rate of patients. If we do that, we are able to meet demand for service at a low cost, yet with acceptable delay times and acceptable blocking rates. But in spite of these patterns, it was common practice in the US, for many years, to determine hospital bed capacity by using the mean bed occupancy measure. This was done both by policy-makers and various levels of hospital management. Green [20] showed by simple M/M/S queueing model that this method was wrong; in Europe, Harper and Shahani [26] claiming the same, developed a simulation tool, for fitting acceptable bed occupancy to the monthly and daily arrival-rate patterns; the tool is based on the Length-of-Stay (LoS) statistics, and the refusal rates. In Israel, bed capacities are determined by turnover rates per bed, and the forecasting of future mean LoS. As explained, the long- and short-term analysis of the beds requirements, are helping to deter- mine the necessary bed allocation. Hospitals distinguish between maximal bed capacity and regular bed capacity. The former is the true physical constraint of the system, while the system is designed to operate with the latter. Naturally, the maximal bed capacity itself is mostly fixed (in scale of months), therefore, the available capacity of the MU is determined by more flexible elements such as the number of doctors and nurses.
1.2.2 Managing work-force capacity
The work-force of a hospital includes nurses, doctors, laboratory workers and others. Most of these human resources need very long and expensive training, and together contribute as much as 70% of the hospital expenses. Nursing salaries make up the largest single element in hospital costs [46]. Thus, much attention is needed in managing the work-force capacity.
10 At the top of the work-force planning hierarchy, a long-term staffing problem is solved to ensure that monthly staffing requirements are met. The problem is usually considered at the management level, considering costs and the annual rate of personnel turnover, which reflects dissatisfaction, differences in workload between wards, and seasonal variation in admission rates. Hospital staffing involves determining the number of personnel of the required skills in order to meet predicted requirements. It is sometimes referred to as nurse budgeting, or workforce scheduling in other personnel planning environments. Burke et al. [8] reviewed some articles on the subject. One can determine by queueing models how many nurses should be available to serve patients over a given time slot. The staffing levels can vary between shifts or months, and track the predictable variations in the arrival rates of patients. But so far, much more robust structures are used; in 2004 in the US, the California Department of Health Services (CDHS) published a law that specifies a nurse-to-patient ratios that determine, the minimal staffing levels allowed [43]. In other countries, such as Israel, staffing levels are determined by labor agreements. We will specify only the last development in the field of Health-Care staffing models; in 2006, Jennings and V´ericourt [29] used a queueing model, to develop new nurse-to-patient ratios, that are a function of the MU size. These ratios were developed in the QED regime in order to balance the work-force efficiency and the quality of care. The next stage is to determine each nurse’s shifts using scheduling models. This planning stage is often referred to in the literature as the Nurse Rostering Problem (NRP) or the Nurse Scheduling Problem (NSP). Cheang et al. [11] defined the NRP as a procedure which involves producing a periodic (weekly, fortnightly, or monthly) duty roster for nursing staff. The schedules are often restricted by legal regulations, personnel policies, nurses’ preferences and many other requirements that may be hospital-specific. It can be quite complex. Naturally, one of these constraints is the minimal staffing level needed to satisfy the service standards, calculated in the previous stage. There are a few reviews of the different methods for NRP, the most recent being those of Cheang et al. [11] and Burke et al. [8]. There are also some general survey papers in the area of personnel rostering such as that of Ernst et al. [15]. After the scheduling phase comes the third step, the lowest level of the hierarchy: the reallocation of nurses. This phase is a fine-tuning of staffing and scheduling. It involves determining how float nurses are assigned to units based on nonforecastable changes or absenteeism. See, for example, Bard and Purnomo [4].
11 1.3 The QED (Quality- and Efficiency-Driven) regime
As opposed to the hierarchy noted above, we suggest a unified method that will determine the bed allocation and nurse staffing levels simultaneously, in the QED regime. We shall focus on QED queues in order to balance patients clinical needs for timely service with the economical preferences of the system to operate at maximal efficiency. We will start with a short description of the QED regime, and its relevance to our environment, as described below by Mandelbaum [34].
1.3.1 Introduction to the Halfin-Whitt (QED) Regime
Many real-life service systems can be operationally represented by queueing models. In such systems, a particular operational regime often takes place, under different circumstances: it is characterized by high levels of resource-utilization jointly with short periods of queueing delays, the latter being one order of magnitude shorter than service durations. For example, telephone agents in call centers are usually heavily utilized, the service time is naturally measured in minutes, and average waiting times in well-managed enterprises are in the order of seconds. In urban transportation, average parking time is several hours while the search for a parking spot is naturally measured in minutes. Finally, in health care, a patient can wait several hours in the EW prior to being assigned to a MU, where the LoS is measured in days. The modus operandi described above is typical of so-called QED (Quality- and Efficiency- Driven) queues. These have recently enjoyed considerable attention in queueing research, both from the theoretical and the applied points of view [3, 17]. Broadly speaking, the QED regime enjoys the following operational characteristics:
• QED queues are many-server queues. (Formally, the number of servers converges to infinity. Importantly, however, QED-based approximations turn out superb for moderate and even small-size systems [51].)
• QED queues are characterized by high service-quality. As mentioned, waiting times have an order of magnitude smaller than service times. Moreover, a significant percentage of the customers (e.g. 30 – 70%) get served immediately upon arrival; abandonment probabilities, in systems with impatient customers, are small.
• A standard measure of service-efficiency is servers’ utilization (say, the fraction of time that telephone agents answer calls or the percentage of occupied beds in a hospital ward). QED queues are characterized by high servers’ utilization. (Theoretically, utilization converges to
12 100% as the number of servers increases indefinitely; this is achieved by delicately balancing service workload with capacity, which gives rise to high levels of both service Quality and Efficiency - hence the term QED.)
• QED queues adhere to some version of the so-called square-root staffing rule. For example, consider a queueing system with a single type of customer and define its offered load by R = λ · E[S], where λ is the arrival rate and E[S] is the mean service time. Then QED staffing corresponds to a number of servers n that is given by √ n ≈ R + β R, (1.1)
where the QoS (Quality-of-Service) parameter β can be either positive or unrestricted, de- pending on the model at hand.
The square-root staffing rule (1.1) was described already by Erlang [14], as early as 1924. (He reported that it had in fact been in use at the Copenhagen Telephone Company since 1913.) However, its formal analysis awaited the seminal paper by Halfin and Whitt [23], in 1981. Halfin and Whitt [23], in the context of the Erlang-C (M/M/n) queue and its generalization GI/M/n, proved the following key result: as n increases indefinitely, sustaining the QED staffing (1.1) with some β > 0 is equivalent to the (steady-state) delay probability converging to some α, 0 < α < 1; (α and β are 1-1 related). An easy consequence of [23] and classical Erlang-C formula is √ that the mean waiting time is of the order 1/ n, which justifies our prior remark on service quality: if, say, n = 100 then the mean waiting time will be in the order of 0.1 × E[S]. In this research proposal we concentrate mainly on QED queues in Health-Care systems, more specifically in hospitals.
1.3.2 QED queues in hospitals
The only work that viewed hospital queues in the QED regime is that of Jennings and V´ericourt [29]. They analyzed the prevalent staffing practice of an a priori-fixed patient-to-nurse ratio (for example, 6-children-per-nurse in a pediatric ward). They showed that such a practice results in either over- or under-staffing in large or small MUs respectively, which can be remedied by square-root staffing. Their mathematical framework [28] is a special Jackson closed-network (machine-repairmen) model of the MU, where the circulating customers are patients’ requests for nursing assistance. (Randhawa and Kumar [45] is a related model, where losses replace the delays in [29, 28].) Jennings and V´ericourt[29, 28] considered the MU model as depicted in Figure2. It is, in fact, a M/M/s/∞/n queueing model. Specifically, there are n beds, all occupied by patients. From time to
13 The medical Center:
Services MU1
MU2
Emergency Patient Arrivals Ward MU3 discharge
Blocked Patient MUn patients discharge time, these patients require the assistance of one of s nurses, in which case we refer to the patients’ state as needy . Otherwise, their state is dormant. The state of patients alternate between needy and dormant states. When patients become needy and an idle nurse is available, they are immediately treated by a nurse. Otherwise, patients wait for an available nurse. The queueing policy is First Come FirstJennings Served (2006): (FCFS). It is assumed that treatment times, i.e. needy-state times, are i.i.d. exponential with rate µ; dormant times are also i.i.d. exponential with rate λ. It is also assumed that the needy and dormant times are independent of each other.
s n-s Needy exp( μ )
n Dormant exp( λ )
Figure 2: Jennings and V´ericourt’smodel [29, 28]
Jennings and V´ericourtdefine the operation regimes for that system as follows: Let us define sn
sn λ as the number of nurses in the n-th system,s ¯ = limn→∞ n , and r = λ+µ then
• Ifs ¯ < r, the system operates in an Efficiency Driven (ED) staffing regime (T > 0)
• Ifs ¯ = r, the system operates in a QED staffing regime (T ≥ 0 and small)
• Ifs ¯ > r, the system operates in a Quality Driven (QD) staffing regime (T = 0) where T is a fixed parameter, representing the required time for service. Then the appropriate √ QED staffing rule is: sn = drn + β ne. Naturally, the QED limit of the probability of delay was calculated, and a central result of their article [28] is their
Proposition 1. The approximate probability of delay has a nondegenerate limit α ∈ √ sn (0, 1) if and only if βn = n − r n → β, as n → ∞, for some β ∈ (−∞, ∞), with −1 β 2 Φ √ −β √ rr¯ α = 1 + e r2 r . Φ −√β r r¯
Herer ¯ := 1 − r.
14 Significantly, the QED regime is many-server asymptotic, as the number of servers increases indefi- nitely. Yet it is also relevant for application in small systems, including nurse staffing, being able to accommodate a small number of nurses (single-digit and above). This relevance is a consequence of the surprising accuracy of square-root staffing, a fact discovered in [6] and also recently supported by the results of Jennings and V´ericourt[29]. As mentioned before, the QED regime arises naturally as the mathematical framework for From the point of view of the beds and patients: patient-flows from the EW to the MUs. Indeed, consider the queueing times at the EW (resulting in
MU hospitalization) vs. the consequent Length-of-Stay (LoS) at the MUs: hours vs. days is typical.
Also, the number of beds (servers) in MUs of moderate-to-large hospitals is in the 10’s (35-50 beds in each of 5 MUs, at the Technion affiliated hospital)Needy - which is well within the accuracy limits of
1-p QED asymptotics. 1 Arrivals 3
1.3.3 QED queues in our Internal-Ward applicationp Patient discharge, Bed in preparation
In this research, we will first extend theDormant Nurse-To-Patient model of Jennings and V´ericourt[29] to accommodate jointly bed allocation and2 nurse staffing, in the QED regime. Bed allocation Blocked determines blockingpatients probabilities of the MUs; nurse staffing determines the delay probabilities of patients waiting for medical care inside the MUs. The combination of these two issues will allow us to determine the appropriate capacity planning while gaining a deeper understanding of the relationship between bed allocation and nurse staffing, which are usually considered separately.
N beds
Patient is Needy Arrivals 1-p from 1 3 the EW p Patient discharge, Bed in Cleaning Patient is Dormant
Blocked 2 patients
Figure 3: The MU model as a semi-open queueing network
In order to do that, in Chapter2, we will model the MU as an semi-open queuing system (see
15 Figure3) and reduce it to a closed Jackson network, as will be shown later (see Figure5), which yields a product-form steady-state. In Chapter3 we define some system measures to this network. These system measures are designed to enable us to answer the following questions:
1. How many nurses should be planned for the unit? One can ask this in the context of either providing reasonable service levels, or from the viewpoint of cost / profit optimization. An answer to this question can be based, for example, on the following measures:
(a) What is the probability of waiting for a nurse?
(b) What is the probability to wait more than T units of time?
2. How many beds should be planned for in the unit? This question could also be answered from the viewpoint of service quality, i.e., providing reasonable availability, or from the viewpoint of cost optimization. Again some measures should be calculated such as:
(a) What is the probability of blocking? i.e., the fraction of time that the system is in full capacity, which translates into the percentage of patients not admitted upon arrival.
One should notice that the blocked patients are getting stuck in the EW which, in turn, can result in reducing the available capacity of the EW itself, as well as hurting patients’ safety and well-being.
Naturally, one would like the answers for Questions1 and2 to be synchronized. Next, in Chapter4, we define QED scaling for the system in Figure3 in the following way: s λ λ √ s = + β + o( λ), −∞ < β < ∞ (i) (1 − p)µ (1 − p)µ s pλ λ pλ λ √ n − s = + + η + + o( λ) −∞ < η < ∞ (ii) (1 − p)δ γ (1 − p)δ γ where p,µ,δ, and γ are fixed model parameters. Term (ii) corresponds to requests queueing for a nurse, and Term (i) corresponds to the effective capacity in the non-queue states. Now, the QED limits of our performance measures can be calculated. For example, as λ, s and n increase indefinitely and simultaneously, according to the above QED scalings, and β 6= 0, then −1 R β q δγ −∞ Φ η + (β − t) µ(pγ+(1−p)δ) dΦ(t) lim P (W > 0) = 1 + √ , λ→∞ φ(β)Φ(η) φ( η2+β2) 1 2 2 η1 β − β e Φ(η1)
q µ(pγ+(1−p)δ) where W denotes waiting for nurse-service, η1 = η − β δγ ; φ(·) and Φ(·) are the standard normal density and distribution functions, respectively. This limit and a few more QED limits are
16 proved in Chapter5. The result supports our definition of the QED regime (non-degenerate delay probability).
1.4 Other related models
Our model in closely related to the one in Khudyakov [32], where it was developed for a call center with an Interactive Voice Response (IVR) system. In fact, all our heavy-traffic approximations have the same structure as those in [32]. With this observation as a starting point, we introduce in Chapter 8 a generalization that covers both systems, as well as some additional modeling possibilities of our MU system. Note that Khudyakov’s model is general, in the sense that it covers various other models such as the M/M/S/S loss system, M/M/N/N (Erlang-B), M/M/S/N, and M/M/S (Erlang-C).
1.5 Other operating regimes
It is acceptable to distinguish between three operating regimes: ED, QED and QD. The QED regime was explained earlier. We will now discuss briefly the ED and QD regimes, and our comments apply to systems with a moderate to a large number of servers.
1. Quality-Driven (QD) Regime. QD queues are characterized by high service-quality and low utilization. Specifically, a large majority of the customers (e.g. 70 – 100%) recives service immediately upon arrival, and servers’ utilization relativly low (e.g. less than 80%).
2. Efficiency-Driven (ED) Regime. ED queues are characterized by high servers’ utilization, with low service-quality. Essentially all customers are delayed proir to service and utilization is close to 100%.
Naturally, the ED regime is mostly useful when staffing of very expensive servers is considered, for example those doctors which have very long training periods and high salaries. On the other hand, the QD regime is useful when extremely “expensive” customers are on hand; for example, if there is a very delicate and expensive machine, or if there is a bottleneck-machine in the factory, then the factory does not want it to stop working; they would allocate a special team to operate this machine, even if these workers are partly unoccupied. Consider again a queueing system with a single type of customer, in which the offered load is R = λ · E[S], where λ is the arrival rate and E[S] is the mean service time. Then QD staffing corresponds to the number of servers n given by
n ≈ R + δR, δ > 0,
17 which yields over-staffing with respect to the offered-load, and ED staffing corresponds to the number of servers n given by
n ≈ R − γR, 0 < γ < 1, which yields under-staffing with respect to the offered-load.
1.6 Research objectives
There are several reasons why we decided to concentrate on capacity management. First, the main resource used in Health-Care systems, as in many other service industries, is the human resource. Doctors, nurses, therapists, laboratory technicians, and so forth are the main resources of that system and their salaries constitute 70% of hospital expenditure [42]. The second reason is that these personnel have very long training periods, and Health-Care systems suffer from a chronical shortage of medical personnel, which has vary bad impact on the Health-Care system. For example, in the US alone, in 2005 there were 1.1 million FTE (Full Time Euivalent) Registrated Nurse (RN) jobs [1], but there is still a chronic shortage of nurses; the American Hospital Assosiation (AHA) reported that US hospitals had an estimated 116,000 RN vacancies as of December 2006 [2], and that the personnel shortage causes some very serious problems in the majority of hospitals, such as decreased staff satisfaction (in 49% of the hospitals), EW overcrowding (36%), diverted EW patients (35%), reduced number of staffed beds (17%), and increased waiting times to surgery (13%). Hence, we seek to develop optimal capacity management policies for Health-Care systems, using stochastic processes. We intend to develop strategies (via asymptotic constraint satisfaction or cost/profit optimization) for joint nurse-staffing and bed-allocation. Our service-level objectives will reflect blocking phenomena and the response time of the medical staff. We would like validate such models using real patient-track data from RFID (Radio Frequency IDentification) systems, used in hospitals. In the future, we would like to combine as many as possible psychological and managerial issues into our models. The model we describe in this work has natural extensions, such as multi-classes of patients, additional phases of clinical treatment, adding doctors (most likely working in the ED regime, in parallel to nurses in the QED regime), time-variability and random parameters, and more. These suggestions and more will be discussed in Chapter 11. Hopefully, once we obtain appropriate hospital data, they will be used to help one determine those extensions, if any, which yield the best model validity. The rest of this document is organized as follows: the first model we are going to develop is introduced in Section2. System measures will be defined in Section3. The QED regime of the
18 system is described in Section4. The development of heavy traffic limits, in the QED regime, of our system-measures are detailed in Section5. In Section6 we compare the approximations with the exact calculations, trying to define the ranges where the approximation is most accurate. In Section 7 we compare our model with other queueing models investigated in the past. A generalization of our model and the appropriate QED asymptotics are presented in Section8. Fluid limits of a simpler model are described in Section9. A method for defining Optimal Design is shown in Section 10. Finally, suggestions for Further Research are discussed in Section 11.
19 2 Extended Nurse-to-Patient model
2.1 The medical unit: Internal Ward (IW)
We consider the following medical unit: the maximal number of beds available is n and the number of nurses serving patients in the unit is s. Typically, the number of nurses is fewer than the number of beds, i.e. s ≤ n, which is assumed here as well. Patients in the unit require the assistance of a nurse from time to time. When such assistance is required we refer to the patient’s state as needy. Otherwise, we call the patient’s state dormant. When patients arrive, they start in a needy state and then alternate between needy and dormant states. When patients are discharged from the hospital, they leave from the needy state. The last treatment can thus reflect the discharge process. After a patient leaves that medical unit, his bed needs to be made available for a new patient. This is usually done by a cleaning crew and not by the nurses of the unit. We will refer to that state of a bed as cleaning. When patients become needy and an idle nurse is available, they are immediately treated by a nurse. Otherwise, patients wait for an available nurse. The queueing policy is FCFS (First Come First Served). After completing treatment, a patient is discharged from the hospital with probability 1−p or goes back to a dormant state with probability p until additional care procedures are required. We assume that the treatment times, are independent and identically distributed (i.i.d.) as an exponential random variable with rate µ and that the dormant times are also i.i.d. exponential with rate δ. As noted above, after the discharge process the bedding must be changed. We assume that cleaning times are i.i.d. exponential with rate σ. We also assume that the needy, dormant, and cleaning times are independent of each other and of the arrival process. The arrival process is assumed to be a Poisson process at rate λ. Another major assumption concerning the arrivals is that if patients arrive at the MU in order to be hospitalized but the unit is full, they are diverted elsewhere, for example back to the Emergency Ward (EW) or to other units of the hospital. Thus, one can view this as blocking of the unit; in this situation we say that the MU is blocked and the request is lost. (In a call center such a situation corresponds to customers encountering a busy signal). To this end, the MU is modeled as the semi-open queueing network. For reading convenience, we reconstruct the IW model form Figure3, here as well.
20 From the point of view of the beds and patients:
Needy
1-p 1 Arrivals 3
p Patient discharge, Bed in preparation
Dormant
2 Blocked patients
N beds
Patient is Needy Arrivals 1-p from 1 3 the EW p Patient discharge, Bed in Cleaning Patient is Dormant
Blocked 2 patients
Figure 4: The IW model as a semi-open queueing network
2.2 The model
The analysis of the above semi-open network can be reduced to that of a closed Jackson network1, which yields a product-form steady-state. A scheme of our model in this form appears in Figure5. This is done by representing our model of the medical unit as a system with four nodes. Node 1 represents beds with patients in a needy state. Node 2 represents beds with patients in a dormant state. For convenience, we sometimes refer to a bed with a patient as simply a patient. Node 3 represents beds in preparation, i.e. in a cleaning state. Node 4 represents prepared beds, awaiting a patient. Nodes 1 to 3 are all multi-server queues. The first node can handle, at most, s patients at one time. The second and third nodes can “handle” or contain at most n patients at a time. Node 4, which is a single-server queue, represents the external arrival process into the unit as will be explained later. Let Q(t) = (N(t),D(t),C(t)) represent the number of beds in the needy, dormant or cleaning
1A Jackson network consists of several interconnected queues; it contains an arbitrary but finite number N of service centers, each has an infinite queue. Let i and j denote service stations in that network. The service discipline is FCFS, where the service time in station i is drawn independently from the distribution exp(µi); (we could have state dependent service rates µi(ni)). Customers travel through the network according to transition probabilities.
Thus, a customer departing from station i chooses the queue in station j next with probability Pij . All the customers are identical; they all follow the same rules of behavior. If the network is open then the arrivals from outside to the network (source) arrive as a Poisson stream with rate λ, and from each node there is at least one path to exit, i.e. the probability that a customer entering the network will ultimately depart from the network is 1. If the network is closed, there are no arrivals or departures hence, there is a constant population of customers in the network.
21 Our Model:
exp(λ ),1 exp(μ ), S
4 1
exp(δ ), N
p 2
1-p exp(γ ), N
3
Figure 5: The IW model as a closed Jackson network states respectively.From the point Since ofn isview the of maximum the patient: number of patients/beds in the system, i.e. needy, dormant or in cleaning, then N(t) + D(t) + C(t) Needy≤ n, for all t ≥ 0. The process Q is a finite-state 1-p continuous-timeArrivals Markov chain. The states of the chain1 will be denotedPatient by the discharge triplets {(i, j, k)|i + j + k ≤ n, i, j, k ≥ 0}. A state (i, j, k) represents a situation where i needy patients are being served p or wait for service, j dormant patients are in the unit but need no service at the time, and k beds are being prepared for future patients. iDormant+ j + k ≤ n. Our medical unit model can be viewed as part of a closed Jackson network. The first node Blocked 2 (needy) can be modeledpatients as s servers with a queue in front of them, with an exponential service time at a rate of µ. The second node (dormant) is an infinite-server node, with an exponential service time at a rate of δ. The third node (cleaning) is also an infinite-server node, with an exponential service time at rate of γ. A new patient can enter the unit only if N(t) + D(t) + C(t) < n. Thus, the process of admitting new patients into the unit has the intensity: λ if i + j + k < n, λ(i, j, k) = 0 otherwise.
In order to formulate this situation a fourth node has been added in which we have a single exponential server of rate λ. Generally, this type of a closed Jackson network has the following product form solution for its stationary distribution [19]:
π1(i)π2(j)π3(k)π4(l) P π1(a)π2(b)π3(c)π4(d) , i + j + k + l = n, π0(i, j, k, l) = a+b+c+d=n 0 , otherwise.
22 Here πm(i) is the steady state probability for node m, m = 1, 2, 3, 4 (M/M/s, M/M/∞, M/M/∞, M/M/1 respectively). The stationary probability π(i, j, k) of having i needy patients, j dormant patients and k beds in cleaning can thus be written in a product form as follows: i j k 1 λ 1 pλ 1 λ π0 , 0 ≤ i + j + k ≤ n, π(i, j, k) = ν(i) (1−p)µ j! (1−p)δ k! γ 0 , otherwise.
Here ν(i) is defined as i! , i ≤ s, ν(i) := s!si−s , i ≥ s, where π0 is given by (see AppendixA)
i j k X 1 λ 1 pλ 1 λ π−1 = 0 ν(i) (1 − p)µ j! (1 − p)δ k! γ 0≤i+j+k≤n n l X 1 λ pλ λ = + + l! (1 − p)µ (1 − p)δ γ l=0 n l m i X X X 1 1 1 λ + − · s!si−s i! (m − i)!(l − m)! (1 − p)µ l=s+1 m=s+1 i=s+1 pλ m−i λl−m . (2.1) (1 − p)δ γ Note that π is also a function of n and s. In order to emphasize this dependence, we shall sometimes use πn(·), πn,s(·), etc. In this work we would like to focus on some managerial questions such as:
1. How many nurses should be planned for the unit? One can ask this in the context of providing reasonable service levels, or from the viewpoint of cost / profit optimization. An answer to this question can be based, for example, on the following measures:
(a) What is the probability of waiting for a nurse?
(b) What is the probability of waiting more than T units of time?
2. How many beds should be planned for in the unit? This question could also be answered from the viewpoint of service quality, i.e., providing reasonable availability, or from the aspect of cost optimization. Again some measures should be calculated such as:
(a) What is the probability of blocking? i.e., the amount of time when the system is in full capacity (i + j + k = n), which translates into the percentage of patients not admitted into the MU.
23 Naturally, one would like the answers for Questions1 and2 to be synchronized.
2.3 Alternative models
In the following chapters we will conduct a stationary analysis of the above stated model. Nev- ertheless, we also suggest alternative ways to model the system, which we will use later for our non-stationary analysis. In this chapter we defined several additional possibilities to model the system. These models are illustrated by figures that show only the dormant and needy states (i.e. without cleaning). In the following subsections, Qi will always represent the number of patients at node i.
2.3.1 Proposal 1
In this first proposal, we assume that the arrival rate into the IW is linearly related to the occupancy level of the ward; if the occupancy increases, the arrival rate decreases. This assumption capture various effects: First, when total load is normal, if the hospital operates in a parallel setting, (i.e. there are a few identical IWs in the hospital), there is usually someone that balances the system by transferring patients among wards. Second, when the total load is high, there are balancing effects that reduce arrival rates, such as diversions to other hospitals, and doctors that refrain from referring additional patient during over-loaded periods. The system is presented in Figure6 in two alternative versions. We regard the situation as a closed network, with a state-dependent arrival rate, in which λ(Q) = λ · (n − Q1 − Q2), where Q1 is the number of patients in the needy state, Q2 is the number of dormant patients, and n is the number of beds in the ward.
2.3.2 Proposal 2
In this possibility, the arrival rate is fixed as long as there is a bed available in the system. This is equivalent to the model presented in section 2.2, but without the cleaning state. The system is presented in Figure7.
2.3.3 Proposal 3
We propose another model that is presented in Figure8. Here we relax the constraint on n and ask the following question: What should s and n be so that the probability of waiting is less than α and the probability of exceeding n is less than β, where the interpretation of having more than n beds
24
Alternative models: exp(μ ), S
μμn ()QSQ=∧ ( ) exp(μ ), S 1
μμn ()QSQ=∧ ( ) Poiss()λ 1 1-p 1 n λλ()QnQQ=−− (1 12 ) p exp(μ ), S n exp(δ ),∞ n μμ()QSQ=∧ (1 ) exp(δ ),∞ δ n ()QQ= δ 2 Poiss()λ n 1-p δ ()QQ= δ 2 1 2 n λλ()QnQQ=−− (12 ) p p 2 1-p exp(δ ),∞ exp(λ ),∞ n n δ ()QQ= δ λλ()QnQQ=−− ( ) 2 12
3 2 exp(μ ), S
n μμ()QSQ=∧ ( ) Version 1 Version1 2 Poiss()λ Closed Jackson Network exp(1-pμ ), S n 1 n λ = λn μμ()QSQ=∧ ( ) 1
Poiss()λ p 1-p 1 Figure 6: Alternativeλ n = λ modeln - Proposal 1
exp(δ ),∞ p
n δ ()QQ= δ 2 exp(δ ),∞
δ n ()QQ= δ 2 2
2
exp(μ ), S exp(μ ), S
n n μμ()QSQ=∧ (1 )μμ()QSQ=∧ (1 )
Poiss()λ Poiss()λ 1-p n 1-p 1 n λλ= nI 1 {0}nQ−−12 Q > λλ= nI{0}nQ−− Q > 12 p p exp(δ ),∞
exp(δ ),∞ δ n ()QQ= δ 2 n δ ()QQ= δ 2 2
2
Figure 7: Alternative model - Proposal 2
25
exp(μ ), S
n μμ()QSQ=∧ (1 )
Poiss()λ 1-p 1 n λλ()QnQQ=−− (12 ) p
exp(δ ),∞ is that patients are attended to in the hospital corridors, for example. This is a realistic scenario,
δ n ()QQ= δ since in spite of the fact that bed-allocation is considered as2 static-capacity, there is some flexibility
in managing this resource; in time of need, one can2 add beds in rooms and corridors. In addition,
this alternative might be easier to solve. In this way µ is state dependent but λ is not, and the
system is purely open.
exp(μ ), S
n μμ()QSQ=∧ (1 )
Poiss()λ 1-p 1 λ n = λn
p
exp(δ ),∞
n δ ()QQ= δ 2
2
Figure 8: Alternativeexp( modelμ ), S - Proposal 3
μμn ()QSQ=∧ ( ) 1 Poiss()λ 1-p 2.3.4 Proposal 4 1 λλn = nI {0}nQ−−12 Q > In this model, which is presented in Figure9, we separated the LoSp of patients from the service inside the medical department. The LoS is assumed to be exponential with mean 1/ν and the size exp(δ ),∞
of the population is restricted to n. When then patient is in the system he alternates between the δ ()QQ= δ 2 dormant and the needy states.
2
26
Hospitalization: exp(ν ),∞
n νν()QQ= 2 Arrivals:() Poiss λ n + 1 λλ()QnQ=− ( ) Departure 1
Needy: exp(μ ), S
n μμ()QSQ=∧ (2 )
2
Q1
Dormant : exp(δ ),∞
δδn ()QQQ=− ( )+ 12
3
Figure 9: Alternative model - Proposal 4
exp(μ ), S exp(γ ),∞
n n μμ()QSQ=∧ (1 ) γ ()QQ= γ 3 Poiss ()λ 1-p 1 3 λλn ()QnQQQ=−−− ( ) 123 p
exp(δ ),∞
δ n ()QQ= δ 2
2
27
3 System measures
We now return to the first model, as stated in Chapter 2.2, and our further analysis is aimed exclusively at this.
3.1 Probability of blocking
From the stationary probability, we will now deduce the probability Pl that there are l beds occupied in the system (0 ≤ l ≤ n). The beds could be occupied by patients in needy or dormant states or in the cleaning state. We will use the following relation
l l−i X X X Pl := π(i, j, k) = π(i, j, l − i − j). i,j,k≥0 i=0 j=0 i+j+k=l We distinguish two cases:
1. l ≤ s:
l l−i i j l−i−j X X 1 λ 1 pλ 1 λ P = π l 0 i! (1 − p)µ j! (1 − p)δ (l − i − j)! γ i=0 j=0 1 λ pλ λl = π + + 0 l! (1 − p)µ (1 − p)δ γ
2. l > s: i j l−i−j Pl Pl−i 1 λ 1 pλ 1 λ Pl = π0 i=0 j=0 ν(i) (1−p)µ j! (1−p)δ (l−i−j)! γ i j l−i−j Ps Pl−i 1 λ 1 pλ 1 λ = π0 i=0 j=0 i! (1−p)µ j! (1−p)δ (l−i−j)! γ i j l−i−j Pl Pl−i 1 λ 1 pλ 1 λ + i=s+1 j=0 s!si−s (1−p)µ j! (1−p)δ (l−i−j)! γ l 1 λ pλ λ = π0 l! (1−p)µ + (1−p)δ + γ i j l−i−j Pl Pl−i 1 1 λ 1 pλ 1 λ + i=s+1 j=0 s!si−s − i! (1−p)µ j! (1−p)δ (l−i−j)! γ
Thus,
1 λ pλ λl P = π + + l 0 l! (1 − p)µ (1 − p)δ γ (3.1) l l−i i j l−i−j X X 1 1 λ 1 pλ 1 λ + I − , {l>s} s!si−s i! (1 − p)µ j! (1 − p)δ (l − i − j)! γ i=s+1 j=0 where I{l>s} is the indicator function.
One can derive from that expression the quantity Pn, which is the probability of blocking of the medical unit. Pn will also be indicated as P (blocked).
28 3.2 Probability of waiting more than t units of time and the expected waiting time
One of the important parameters of the level of service, is the time spent in-queue. This is the time that a patient may have to wait to be treated. If a patient becomes needy when there are already i other needy patients in the unit, he will need to wait an in-queue random waiting time that follows an Erlang distribution with (i − s + 1)+ stages, each with rate sµ. The probability −sµt Pi−s j that this Erlang-distributed random variable is greater than t is e j=0(sµt) /(j!). Clearly, the patient only waits if i ≥ s. Let W denote the steady state, in-queue waiting time for a hypothetical patient, who just become needy, and denote pn,s(t) as the tail of the steady state distribution of W , given n beds and s nurses. Formally, pn,s(t) = P (W > t). As a consequence of dealing with a closed system, the total activation rate, i.e. the rate at which the collective stable patient population produces needy patients, is modulated by the state of the system, i.e., by the number of needy, dormant and cleaning beds. In addition, in order to calculate the tail of the steady state distribution of W , we need to use the Arrival Theorem for closed networks, quoted here from Chen and Yao [12].
The Arrival Theorem. In a closed Jackson network, the arrival at (or the departure from) any node observes time averages, with the job itself excluded. In particular, the 2 probability that the network is in state x − ei immediately before an arrival (or imme- diately after a departure) epoch at node i is equal to the ergodic distribution, of a closed
network with one fewer job, in state x − ei.
A Let π (x − ei), denote the probability that the system is in state x − ei at the arrival epoch of a customer to node i. Thus, immediately after the arrival of a customer to node i, the state is x. Then the arriving customer sees before him the state x − ei, which corresponds to a network A with one fewer job. Then by the arrival theorem, we conclude that π (x − ei) = πn−1(x − ei). In A particular for the needy state (node 1), π (x − e1) = πn−1(x − e1) = πn−1(i − 1, j, k). The probability that a patient will get service immediately as he become needy is the sum of probabilities that the customer arriving at the needy state will see fewer than s needy patients; by the former notations it is equal to:
n−1 l min{m,s−1} X X X P (W = 0) = πA(i, m − i, l − m). (3.2) l=0 m=0 i=0
2 [12] refers to state x, rather than x − ei as we do; we believe [12] has a typo.
29 The distribution function of the waiting time is:
n−1 X P (W ≤ t) = P (W = 0) + P (there are (i − s + 1) patients who ended i=s their service on time ≤ t|Arrival at the needy state found i needy patients)·
· πA(i, m − i, l − m) =
n−1 l m X X X Z t µs(µsx)i−s = P (W = 0) + πA(i, m − i, l − m) e−µsxdx (i − s)! l=s m=s i=s 0 n−1 l m i−s X X X X (µst)h = P (W = 0) + πA(i, m − i, l − m)(1 − e−µst) (3.3) h! l=s m=s i=s h=0 n−1 l min{m,s−1} n−1 l m X X X X X X = πA(i, m − i, l − m) + πA(i, m − i, l − m) l=0 m=0 i=0 l=s m=s i=s n−1 l m i−s X X X X (µst)h − πA(i, m − i, l − m) e−µst h! l=s m=s i=s h=0 n−1 l m i−s X X X X (µst)h = 1 − π (i, m − i, l − m) e−µst. n−1 h! l=s m=s i=s h=0 Therefore, the tail steady state distribution of W is
n−1 l m i−s X X X X (µst)h P (W > t) = π (i, m − i, l − m) e−µst (3.4) n−1 h! l=s m=s i=s h=0 and the expected waiting time E[W] can be derived via this tail formula, i.e.,
n−1 l m i−s Z ∞ Z ∞ X X X X (µst)h E[W ] = P (W > t)dt = π (i, m − i, l − m) e−µstdt n−1 h! 0 0 l=s m=s i=s h=0 n−1 l m i−s X X X X Z ∞ (µst)h = π (i, m − i, l − m) e−µstdt n−1 h! m=s 0 l=s i=s h=0 (3.5) n−1 l m i−s X X X X 1 = π (i, m − i, l − m) n−1 µs l=s m=s i=s h=0 n−1 l m 1 X X X = π (i, m − i, l − m)(i − s + 1). µs n−1 l=s m=s i=s This formula is exactly the same as the one found in Gross and Harris [22] pg. 193: In a closed
Jackson network with M/M/cj nodes, the mean waiting time at node j for a network containing n customers is E(W (n)) = 1 Pn−1 (i − c + 1)p (i, n − 1) where p (i, n − 1) is the marginal j µj cj i=cj i j j
30 probability of i in an (n − 1)-customer system at node j. Therefore, for our system
n−1 1 X E(W ) = (i − s + 1)p (i, n − 1) µs 1 i=s n−1 n−1 l 1 X X X = (i − s + 1) π (i, m − i, l − m) (3.6) µs n−1 i=s l=i m=i n−1 l m 1 X X X = (i − s + 1)π (i, m − i, l − m). µs n−1 l=i m=i i=s
3.3 Probability of delay
The probability of delay in terms of previous definitions is P (W > 0). In order to find it we will again use the Arrival Theorem for closed networks, cited earlier on Page 29. Accordingly, we can derive performance measures of a medical unit with n beds and s nurses, by the steady-state distribution of the same system with n−1 beds and s nurses. The probability that a patient who becomes needy has to wait, is the probability that a patient will find more than s needy patients in a system with n beds, and this is exactly the steady-state probability of having more than s needy patients in a system with n − 1 beds. By the notions of the arrival theorem, a patient entering node 1 (as he arrives) sees the system in state x − ei with probability πn−1(x − ei) (no matter where he cames from). After his entrance the system state will be x. Therefore, if we want to know what is the probability that the patient will see the station full, we need to add up the probabilities that in the vector x − e1 the first element x1 − 1 will exceed s patients, i.e., we need to add together the arrival P probabilities of all x such that x1 − 1 ≥ s and |x| = i xi ≤ n . Thus,
X A X A P (W > 0) = π (x − e1) = π (i, j, k)
x||x|≤n;x1−1≥s i,j,k|i+j+k≤n−1,i≥s n−1 l m (3.7) X X X X = πn−1(i, j, k) = πn−1(i, m − i, l − m). i≥s l=s m=s i=s
Thus, for the system with n beds and s nurses, the percentage of patients which are required to wait before being served, coincides with the probability that in a system with n − 1 beds and s nurses, all the nurses are busy. Formally, P (W > 0) = Pn−1(N(∞) ≥ s).
3.4 Average occupancy level
Pn Pl Pm The average occupancy level can be found by OC(n, s) = l=0 m=0 i=0 mπn(i, m − i, l − m).
31 4 The QED regime
Consider a sequence of s-server queues, indexed by n. Let the arrival-rates λn → ∞, as n → ∞, eff λn and fixed µ the service-rate. Define the offered load by Rn = µ . The QED regime is achieved √ by choosing λ and s so that s (1 − ρ ) → β, as n → ∞, for some finite β. Here ρ = Rn . n n n n n sn
When patients have infinite patience, ρn may be interpreted as the long-run servers’ utilization and then one must have 0 < β < ∞. Otherwise, ρn is the offered load per server and −∞ < β < ∞ is allowed. Equivalently, the staffing level is approximately given by
p sn ≈ Rn + β Rn, − ∞ < β < ∞.
eff λn λn λn In our system λn = 1−p , Rn = (1−p)µ , and ρn = (1−p)sµ . Let λ, s and n tend to ∞ simultaneously so that: s λ λ √ s = + β + o( λ), − ∞ < β < ∞, (i) (1 − p)µ (1 − p)µ s s s λ pλ pλ λ λ √ n − s = η + + η + + η + o( λ), (ii) 1 (1 − p)µ (1 − p)δ 2 (1 − p)δ γ 3 γ
− ∞ < η1, η2, η3 < ∞.
First we reduce the number of parameters.
Theorem 1. Let λ, s and n tend to ∞ simultaneously. Then the conditions s λ λ √ s = + β + o( λ), − ∞ < β < ∞, (i) (1 − p)µ (1 − p)µ s s s λ pλ pλ λ λ √ n − s = η + + η + + η + o( λ), (ii) 1 (1 − p)µ (1 − p)δ 2 (1 − p)δ γ 3 γ
− ∞ < η1, η2, η3 < ∞ are equivalent to the conditions s λ λ √ (i) s = + β + o( λ), −∞ < β < ∞ (1 − p)µ (1 − p)µ s (4.1) pλ λ pλ λ √ (ii) n − s = + + η + + o( λ), −∞ < η < ∞ (1 − p)δ γ (1 − p)δ γ
q δγ q γp q (1−p)δ where η = η1 µ(γp+(1−p)δ) + η2 γp+(1−p)δ + η3 γp+(1−p)δ .
32 Proof. Clearly, one can rewrite the second condition in the form s s ! δγ r γp (1 − p)δ n − s = η + η + η 1 µ(γp + (1 − p)δ) 2 γp + (1 − p)δ 3 γp + (1 − p)δ s pλ λ pλ λ √ + + + + o( λ), − ∞ < η , η , η < ∞. (1 − p)δ γ (1 − p)δ γ 1 2 3
q δγ q γp q (1−p)δ Setting η = η1 µ(γp+(1−p)δ) + η2 γp+(1−p)δ + η3 γp+(1−p)δ one obtains (4.1). The first condition is the same. This proves the statement.
We can rewrite the QED condition (4.1) in the following form
pλ λ n − s − − γ lim (1−p)δ = η, −∞ < η < ∞ (i) λ→∞ q pλ λ (1−p)δ + γ √ λ lim s 1 − = β, −∞ < β < ∞ (ii) λ→∞ (1 − p)sµ where the second term defines the situation on the servers (i.e. the effective space in the service station) and the first term defines the effective space remaining in the “non-queue” stations. For convenience we denote
λ pλ λ R = ,R = ,R = , N (1 − p)µ D (1 − p)δ C γ and
λ ρ = . (1 − p)sµ
For some technical reasons we must distinguish between two cases: β = 0 and β 6= 0. This separation results in two separate QED conditions: lim n−√s−RD−RC = η, −∞ < η < ∞, (i) λ→∞ R +R QED = √ D C (4.2) RN limλ→∞ s 1 − s = β, −∞ < β < ∞, β 6= 0, (ii) and n−s−RD−RC limλ→∞ √ = η, −∞ < η < ∞, (i) RD+RC QED0 = √ RN limλ→∞ s 1 − s = β, β = 0, (ii) where µ,p,δ and γ are fixed parameters.
33 5 Heavy traffic limits and asymptotic analysis in the QED regime
In this chapter we develop heavy-traffic approximations of the system-measures introduced in Chap- ter3. As a first stage we present four lemmas; their proofs are in AppendixB.
Lemma 1. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED conditions.
Define ζ1 as the expression
n−s−1 e−RN 1 X 1 ζ = (R )s (R + R )l e−(RD+RC ). 1 s! N 1 − ρ l! D C l=0 Then
φ(β)Φ(η) lim ζ1 = . λ→∞ β
Lemma 2. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED conditions.
Define ζ2 as the expression
n−s−1 −(RN +RD+RC ) n−s l e ρ X 1 RD RC ζ = (R )s + . 2 s! N 1 − ρ l! ρ ρ l=0 Then
p 2 2 φ( η + β ) 1 η2 lim ζ2 = e 2 1 Φ(η1). λ→∞ β
Lemma 3. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED or QED0 conditions. Define ξ as the expression
X 1 ξ = (R )i (R )j (R )k e−(RN +RD+RC ). i!j!k! N D C i,j,k|i≤s, i+j+k≤n−1 Then s ! Z β δγ lim ξ = Φ η + (β − t) dΦ(t). λ→∞ −∞ µ(pγ + (1 − p)δ)
Lemma 4. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED0 conditions. Define ζ as the expression
n−s−1 n−s−k−1 n−s−j−k−1 1 X X 1 X ζ = e−(RN +RD+RC ) R s R jR k ρi. s! N j!k! D C k=0 j=0 i=0 Then s µ(pγ + (1 − p)δ) 1 lim ζ = √ (ηΦ(η) + φ(η)) . λ→∞ δγ 2π
34 5.1 Approximation of the probability of delay
The first approximation will be for the measure: the probability of waiting or the probability of delay. It was defined in Section 3.3, by Formula (3.7).
Theorem 2. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED conditions. Then
R β √ −1 −∞ Φ η + (β − t) B dΦ(t) lim P (W > 0) = 1 + √ λ→∞ φ(β)Φ(η) φ( η2+β2) 1 2 2 η1 β − β e Φ(η1) √ RN δγ −1 where B = = , η1 = η − β B . RC +RD µ(pγ+(1−p)δ)
Proof.
n−1 l m X X X Pn(W > 0) = Pn−1(Q1(∞) ≥ s) = πn−1(i, m − i, l − m) l=s m=s i=s n−1 l m i m−i l−m X X X 1 λ pλ λ = π , 0 s!si−s(m − i)!(l − m)! (1 − p)µ (1 − p)δ γ l=s m=s i=s where
n−1 l X 1 λ pλ λ π = + + 0 l! (1 − p)µ (1 − p)δ γ l=0 −1 n−1 l m i m−i l−m! X X X 1 1 1 λ pλ λ + − . s!si−s i! (m − i)!(l − m)! (1 − p)µ (1 − p)δ γ l=s m=s i=s Thus,
A −1 P (W > 0) = 1 + , n B where
n−1 l X 1 λ pλ λ A = + + l! (1 − p)µ (1 − p)δ γ l=0 n−1 l m i m−i l−m X X X 1 λ pλ λ − i!(m − i)!(l − m)! (1 − p)µ (1 − p)δ γ (5.1) l=s m=s i=s i j k X 1 λ pλ λ = , i!j!k! (1 − p)µ (1 − p)δ γ i,j,k|i≤s, i+j+k≤n−1
35 n−1 l m i m−i l−m X X X 1 1 λ pλ λ B = s!si−s (m − i)!(l − m)! (1 − p)µ (1 − p)δ γ l=s m=s i=s n−s−1 n−s−k−1 n−j−k−1 i j k X X X 1 1 λ pλ λ = s!si−s j!k! (1 − p)µ (1 − p)δ γ k=0 j=0 i=s (5.2) n−s−1 n−s−k−1 n−s−j−k−1 i+s j k X X X 1 1 λ pλ λ = s!si j!k! (1 − p)µ (1 − p)δ γ k=0 j=0 i=0 s n−s−1 n−s−k−1 j k n−s−j−k−1 i 1 λ X X 1 pλ λ X λ = . s! (1 − p)µ j!k! (1 − p)δ γ (1 − p)sµ k=0 j=0 i=0 λ √ Define ρ = (1−p)sµ , then under the QED (part (ii)) assumption that s(1 − ρ) → β, − ∞ < β < ∞, β 6= 0 (of Theorem2) as λ → ∞, we can rewrite the right-hand side in the following way:
s n−s−1 n−s−k−1 j k 1 λ X X 1 pλ λ 1 − ρn−s−j−k B = s! (1 − p)µ j!k! (1 − p)δ γ 1 − ρ k=0 j=0 s n−s−1 n−s−k−1 j k 1 λ 1 X X 1 pλ λ = s! (1 − p)µ 1 − ρ j!k! (1 − p)δ γ k=0 j=0 s n−s−1 n−s−k−1 j k 1 λ ρn−s X X 1 pλ λ − . s! (1 − p)µ 1 − ρ j!k! (1 − p)δρ γρ k=0 j=0 Applying the multinomial theorem yields:
s n−s−1 l 1 λ 1 X 1 pλ λ B = + s! (1 − p)µ 1 − ρ l! (1 − p)δ γ l=0 s n−s−1 l 1 λ ρn−s X 1 pλ λ − + s! (1 − p)µ 1 − ρ l! (1 − p)δρ γρ l=0
= B1 − B2.
λ pλ λ − (1−p)µ + (1−p)δ + γ Multiplying A, B1 and B2 by e we have ξ −1 P (W > 0) = 1 + , ζ1 − ζ2 where ξ, ζ1 and ζ2 where defined in lemmas1-3, and we repeat them for convenience,
i j k pλ X 1 λ pλ λ − λ + + λ ξ = e (1−p)µ (1−p)δ γ , (5.3) i!j!k! (1 − p)µ (1 − p)δ γ i,j,k|i≤s, i+j+k≤n−1 n−s−1 s l pλ 1 λ 1 X 1 pλ λ − λ + + λ ζ = + e (1−p)µ (1−p)δ γ , (5.4) 1 s! (1 − p)µ 1 − ρ l! (1 − p)δ γ l=0 n−s−1 s n−s l pλ 1 λ ρ X 1 pλ λ − λ + + λ ζ = + e (1−p)µ (1−p)δ γ . (5.5) 2 s! (1 − p)µ 1 − ρ l! (1 − p)δρ γρ l=0
36 By Lemmas1,2, and3 if β 6= 0:
φ(β)Φ(η) lim ζ1 = , (5.6) λ→∞ β p 2 2 φ( η + β ) 1 η2 lim ζ2 = e 2 1 Φ(η1), (5.7) λ→∞ β and s ! Z β δγ lim ξ = Φ η + (β − t) dΦ(t), (5.8) λ→∞ −∞ µ(pγ + (1 − p)δ)
q µ(pγ+(1−p)δ) where η1 = η − β δγ . Thus,
−1 R β q δγ −∞ Φ η + (β − t) µ(pγ+(1−p)δ) dΦ(t) lim P (W > 0) = 1 + √ . λ→∞ φ(β)Φ(η) φ( η2+β2) 1 2 2 η1 β − β e Φ(η1) This proves Theorem2.
Theorem 3. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED0 conditions. Then
−1 R 0 q δγ −∞ Φ η − t µ(pγ+(1−p)δ) dΦ(t) lim P (W > 0) = 1 + q µ(pγ+(1−p)δ) λ→∞ √1 (ηΦ(η) + φ(η)) δγ 2π
q µ(pγ+(1−p)δ) where η1 = η − β δγ .
Proof. As before,
A −1 P (W > 0) = 1 + n B where,
i j k X 1 λ pλ λ A = i!j!k! (1 − p)µ (1 − p)δ γ i,j,k|i≤s, i+j+k≤n−1
s n−s−1 n−s−k−1 j k n−s−j−k−1 i 1 λ X X 1 pλ λ X λ B = . s! (1 − p)µ j!k! (1 − p)δ γ (1 − p)sµ k=0 j=0 i=0
−(RN +RD+RC ) λ pλ λ We can multiply each phrase in e where RN = (1−p)µ , RD = (1−p)δ , RC = γ , then
ξ −1 P (W > 0) = 1 + n ζ
37 where, ξ = A · e−(RN +RD+RC ), and ζ = B · e−(RN +RD+RC ). By Lemma3, when β = 0: s ! Z 0 δγ lim ξ = Φ η − t dΦ(t), (5.9) λ→∞ −∞ µ(pγ + (1 − p)δ) and, due to Lemma4: s (1 − p)µ pµ 1 lim ζ = + √ (ηΦ(η) + φ(η)) . (5.10) λ→∞ γ δ 2π Assigning Equations (5.9) and (5.10), we proved Theorem3.
Checking: Is limβ→0 P{β6=0}(W > 0) = P{β=0}(W > 0)? We need to check that: p 1 η2 φ(β)Φ(η) − φ( η2 + β2)e 2 1 Φ(η ) lim 1 β→0 β s (1 − p)µ pµ 1 = + √ (ηΦ(η) + φ(η)) . γ δ 2π
q µ(pγ+(1−p)δ) √ Define: η1 = η − β δγ = η − β C. By L’Hˆopital’srule:
p 1 η2 φ(β)Φ(η) − φ( η2 + β2)e 2 1 Φ(η ) lim 1 β→0 β
1 d p 2 2 η2 = lim φ(β)Φ(η) − φ( η + β )e 2 1 Φ(η1) β→0 dβ p 1 η2 2 2 1 2 1 dφ(β) dφ( η + β ) η2 p 2 2 d(e Φ(η1)) = lim Φ(η) − e 2 1 Φ(η1) − φ( η + β ) β→0 dβ dβ dβ p 1 η2 ! 2 2 1 2 1 1 dφ(β) dφ( η + β ) η2 p 2 2 de dΦ(η1) η2 = lim Φ(η) − e 2 1 Φ(η1) − φ( η + β ) Φ(η1) + e 2 1 β→0 dβ dβ dβ dβ
2 2 2 −β − β −β − η +β 1 η2 = lim Φ(η)√ e 2 − √ e 2 e 2 1 Φ(η1) β→0 2π 2π 1 2 ! p d 2 η1 1 η2 dη1 1 η2 − φ( η2 + β2) e 2 1 Φ(η ) + φ(η )e 2 1 dβ 1 dβ 1
2 2 2 −β − β −β − η +β 1 η2 = lim Φ(η)√ e 2 − √ e 2 e 2 1 Φ(η1) β→0 2π 2π √ 1 √ 1 p 2 2 η2 η2 − φ( η + β ) (−η1 C)e 2 1 Φ(η1) + (− C)φ(η1)e 2 1 √ √ 1 η2 C = φ(η) Ce 2 (ηΦ(η) + φ(η)) = √ (ηΦ(η) + φ(η)) . 2π
5.2 Approximation of the expected waiting time
The second approximation will be for the measure: the expected waiting time. It was defined in Section 3.2, by Formula (3.5). We state the Theorems here, the proofs are in AppendixC.
38 The first theorem gives the approximation for the case where β 6= 0.
Theorem 4. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED conditions. Then √ φ(β)Φ(η) φ( η2+β2) 1 2 β η 1 2 η1 √ 1 √ 1 β β + β e Φ(η1) B − − β lim sE[W ] = √ B λ→∞ µ β √ φ(β)Φ(η) φ( η2+β2) 1 2 R 2 η1 −∞ Φ η + (β − t) B dΦ(t) + β − β e Φ(η1) √ RN δγ −1 where B = = , η1 = η − β B . RC +RD µ(pγ+(1−p)δ)
The second theorem gives the approximation for the case where β = 0.
Theorem 5. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED0 conditions. Then
√ 1 B−1 (η2 + 1)Φ(η) + ηφ(η) lim sE[W ] = √ √ √ λ→∞ 2µ R 0 −1 2π −∞ Φ η − t B dΦ(t) + B (ηΦ(η) + φ(η)) √ RN δγ −1 where B = = , η1 = η − β B . RC +RD µ(pγ+(1−p)δ)
5.3 Approximation of blocking probability
The third approximation will be for the probability of blocking. This measure was defined in Section 3.1, by Formula (3.1). We only state here the approximation theorems, as conjectures supported by our previous experience. We intend to prove these measures in the near future.
Theorem 6. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED conditions. Define B = RN = δγ , then RC +RD µ(pγ+(1−p)δ)
2 p η1 √ νφ(ν )Φ(ν ) + φ( η2 + β2)e 2 Φ(η ) lim sP (block) = 1 2 √ 1 (5.11) λ→∞ β √ φ(β)Φ(η) φ( η2+β2) 1 2 R 2 η1 −∞ Φ η + (β − t) B dΦ(t) + β − β e Φ(η1) √ √ −1 −1 where η = η − √β , ν = √ 1 , ν = η√ B +β , ν = β√ B −η . 1 B 1+B−1 1 1+B−1 2 1+B−1
Theorem 7. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED0 conditions. Define and B = RN = δγ , then RC +RD µ(pγ+(1−p)δ)
1 √ νφ(ν1)Φ(ν2) + √ Φ(η) lim sP (block) = 2π (5.12) 0 √ λ→∞ R Φ η − t B dΦ(t) + √1 √1 (ηΦ(η) + φ(η)) −∞ B 2π where ν = √ 1 , ν = √ η , ν = √ η . 1+B−1 1 1+B 2 1+B−1
39 6 Comparison of approximations and exact calculations
In this section, using some examples, we illustrate the quality of our approximations, when compared against the exact calculations. In the future, we intend to analyze carefully the cases where the approximations work the best, and answer questions such as from which system size (i.e. what n), we can use the approximations. (Formally, these are questions of rates of convergence.) The exact calculations and the approximations were calculated using MATLAB. The first three illustrations are of a small system, where the number of beds is 15. Figures 10-a, 10-b, 10-c differ on one parameter which is p - the probability of staying in the MU after treatment. The parameters are specified in Table1. One can observe that in Figure 10-a the matching between
Figure n λ δ µ p 10-a 15 5 0.25 1 0.25 10-b 15 5 0.25 1 0.5 10-c 15 5 0.25 1 0.75
Table 1: Parameters for small systems the exact calculation and the approximation is almost perfect in the range of interest (i.e. when P (W > 0) ∈ [0.3 − 0.7]). In Figures 10-b and 10-c there is a gap between the two, which means that the approximation is not accurate; we still do not have an explanation for this gap, but we are intending to analyze this carefully in the future.
(a) (b) (c)
Figure 10: Three example of comparison between approximation and exact calculation - Small system
40 The next two illustrations, shown in Figure 11, are of medium systems, where the number of beds is 35 and 50, respectively. Other parameters are specified in Table2. In both cases, one can
Figure n λ δ µ γ p 11-a 35 10 0.25 3 5 0.5 11-b 50 10 0.5 1 10 0.5
Table 2: Parameters for medium systems observe that the matching between the exact calculation and the approximation is very good in the range of interest (i.e. when P (W > 0) ∈ [0.3 − 0.7]).
(a) (b)
Figure 11: Two example of comparison between approximation and exact calculation - Medium system
The next illustration, shown in Figure 12, is of a large system where the number of beds is 90. Other parameters are specified in Table3. In this case too, we observe a good matching between
Figure n λ δ µ γ p 12 90 20 0.25 3 5 0.5
Table 3: Parameters for a large system the exact calculation and the approximation.
41 Figure 12: Comparison of approximation and exact calculation - Large system
The next three illustration-parameters are based on data taken from a survey carried out in Israely Hospital by Marmor [40]. The data we have are partial, not altogether consistent with the data we need. We know that in each of these MUs there are 30 beds, and the average LoS is three days; patients’ average arrival rate is five patients per day. Since we do not have accurate data on service times, we show three options using various assumptions. In all of them, we assume that average cleaning times are one hour. The parameters are specified in Table4 and their related Figure is 13. In all the Figures (13 a-c) we observe a good matching between the exact calculation
Figure n λ δ µ γ p 13-a 30 5/24 4/23 4 1 12/13 13-b 30 5/24 4/21 4 1 36/37 13-c 30 5/24 2/19 2 1 30/31
Table 4: Parameters based on data from Israely Hospital and the approximation in the range of interest (i.e. when P (W > 0) ∈ [0.3 − 0.7]).
42 (a) (b) (c)
Figure 13: Comparison of approximation and exact calculation - Israely Hospital
The last illustration parameters arising from the need to compare Jennings and V´ericourt’s model to ours. They used the following ratio: r = λJ = δ = 0.25. The illustration is shown in λJ +µ δ+µ Figure 14; the parameters are specified in Table5. In this case too, we see a good matching between
Figure n λ δ µ γ p 14 40 10 1 3 1 0.5
Table 5: Parameters based on Jennings and V´ericourt’sarticle the exact calculation and the approximation.
Figure 14: Comparison of approximation and exact calculation - r = 0.25
43 7 Comparison with other models
In this chapter we will present some special cases of our model. We will show that the probability distribution of these models can be represented by our model probability functions, or some other connection between the models.
7.1 The M/M/S/infinity/n system
When λ → ∞, and δ = γ our model will be equivalent to Jennings and V´ericourt’smodel [28]. This is an M/M/S/∞/n system, with exactly n customer in the system, i.e. i + j + k = n. Jennings and V´ericourtused the definition of r = λJ which is equivalent to: r = pδ+(1−p)γ = δ in our λJ +µ pδ+(1−p)γ+µ δ+µ model. As seen before, for each (i, j, k) such that 0 ≤ i + j + k ≤ n,
1 λ i 1 pλ j 1 λk π(i, j, k) = π 0 ν(i) (1 − p)µ j! (1 − p)δ k! δ since λ = ∞ (goes to infinity faster than n and s) the probability of patients being in node 4 is 0, thus, i + j + k = n and π(i, j, k) = π(i, j, n − i − j). We define the marginal distribution π(i, n − i) as j + k = n − i:
X π(i, n − i) = π(i, j, k) j,k|j+k=n−i i j k X 1 λ 1 pλ λ = π 0 ν(i) (1 − p)µ j!k! (1 − p)δ δ j,k|j+k=n−i 1 1 i 1 1 n−i = π (λ)n 0 ν(i) (1 − p)µ (n − i)! (1 − p)δ λ n 1 1 i 1 1n−i = π 0 1 − p ν(i) µ (n − i)! δ λ n 1 1 δ i = π . 0 (1 − p)δ ν(i) (n − i)! µ
Here ν(i) is defined as i! , i ≤ s, ν(i) := s!si−s , i ≥ s.
44 and π0 is given by
i j k X 1 λ 1 pλ 1 λ π−1 = 0 ν(i) (1 − p)µ j! (1 − p)δ k! γ i+j+k=n n i j k X X 1 1 1 p 1 = λn ν(i) (1 − p)µ j!k! (1 − p)δ δ i=0 j,k|j+k=n−i n i n−i X 1 1 1 1 = λn ν(i) (1 − p)µ (n − i)! (1 − p)δ i=0 n s i n−i n i n−i! λ X 1 1 1 1 X 1 1 1 1 = + 1 − p i! µ (n − i)! δ s!si−s µ (n − i)! δ i=0 i=s+1 n n n i n−i! λ 1 1 X 1 1 n! 1 1 = + + − . 1 − p µ δ s!si−s i! (n − i)! µ δ i=s+1 Or in an equivalent form, i n−i 1 1 1 1 π˜0 , i ≤ s, i! µ (n − i)! δ i n−i π(i, n − i) = 1 1 1 1 π˜0 , i > s, s!si−s µ (n − i)! δ 0, otherwise, whereπ ˜0 is given by n n i n−i 1 1 X 1 1 n! 1 1 π˜−1 = + + − . 0 µ δ s!si−s i! (n − i)! µ δ i=s+1 Or in an equivalent form, i n δ π¯0 , i ≤ s, i µ i π(i, n − i) = n i! δ π¯0 , i > s, i s!si−s µ 0, otherwise, whereπ ¯0 is given by
n n i n−i! 1 1 X 1 1 n! 1 1 π¯−1 = n!δn + + − 0 µ δ s!si−s i! (n − i)! µ δ i=s+1 n n i! µ + δ X 1 1 n! δ = n! + − . µ s!si−s i! (n − i)! µ i=s+1 This last version of steady-state probabilities were investigated by Jennings and V´ericourt[28]. In [28] there are no exogenous arrivals (closed system i.e. no λ in our notation), which corresponds to taking λ to infinity at a faster rate than n and s.
45 7.2 Call center with IVR (Interactive Voice Response)
In certain settings, our model will be equivalent to Khudyakov’s model [32]. As seen before, for each (i, j, k) such that 0 ≤ i + j + k ≤ n:
1 λ i 1 pλ j 1 λk π(i, j, k) = π . 0 ν(i) (1 − p)µ j! (1 − p)δ k! γ
We would like to define the marginal distribution π(i, l) as j + k = l. For each (i, l) such as 0 ≤ i + l ≤ n: X π(i, l) = π(i, j, k) j,k|j+k=l i j k X 1 λ 1 pλ λ = π 0 ν(i) (1 − p)µ j!k! (1 − p)δ γ j,k|j+k=l 1 λ i 1 pλ λl = π + . 0 ν(i) (1 − p)µ l! (1 − p)δ γ
Here ν(i) is defined as i! , i ≤ s, ν(i) := s!si−s , i ≥ s, where π0 is given by
i j k X 1 λ 1 pλ 1 λ π−1 = 0 ν(i) (1 − p)µ j! (1 − p)δ k! γ 0≤i+j+k≤n n i j k X X 1 λ 1 pλ 1 λ = ν(i) (1 − p)µ j! (1 − p)δ k! γ i=0 0≤j+k≤n−i n n−i i l X X 1 λ 1 pλ λ = + ν(i) (1 − p)µ l! (1 − p)δ γ i=0 l=0 i l X 1 λ 1 pλ λ = + i! (1 − p)µ l! (1 − p)δ γ i≤s,i+l≤n i l X 1 λ 1 pλ λ + + . s!si−s (1 − p)µ l! (1 − p)δ γ i>s,i+l≤n
These are exactly the steady-state probabilities investigated by Khudyakov [32], in the case: 1 = p + 1 and pK = 1 . For example: if in our model p = 0, our model is comparable θK (1−p)δ γ µK (1−p)µ to Khudyakov’s model with pK = 1.
46 8 Generalizations
In previous chapters, we decided arbitrarily, that patients will start and end their stay at the MU, in service from nurses. One can use other assumptions, which will slightly change the shape of the network. In this chapter, we would like to generalize our findings, and to formulate heavy-traffic approximations that will cover any specific design of flow in the network. In fact, this generalization will also unite our model with the IVR model [32]. The generalization was developed from our experience in developing the approximations in Chapter5. We have not yet proved them, and we might do so if we find it useful for the later stages of our research. This generalization covers any semi-open network, with one service station with s servers, and any finite number of delay procedures. Consider any closed Jackson network that has one M/M/S node (denote as node a), any K finite number of nodes of the type M/M/∞ (denote as node j , j ∈ J = {1, 2, ..., K}) and one node of the type M/M/1. Denote the solution of the balance equations of the steady-state flows of the network
P ρa as ρa and ρj, ∀j ∈ J, and define A = j∈J ρj, B = A . (Explanation: Let λ be the service rate of the M/M/1 node, and P be the transition probability matrix of the network. Define λj as the rate of flow into node j and µ as the rate of flow out of node j, then ρ = λj . The rate λ can be found j j µj j by solving the traffic equations, and should be expressed in terms of λ, and P ). The steady-state probabilities of such networks are: 1 K 1 K i0 Y ij X π0 (ρa) (ρj) , 0 ≤ i + ij ≤ n, ν(i0) ij! π(i0, i1, ..., iK ) = j=1 j=1 0 otherwise.
Here ν(i) is defined as i0! , i0 ≤ s, ν(i0) := i −s s!s 0 , i0 ≥ s, and π0 is the normalization factor, given by
−1 X 1 i0 Y 1 ij π0 = (ρa) (ρj) ν(i0) ij! 0≤i0+i1+...+iK ≤n j∈J n n−i X X 1 i0 1 l = (ρa) (A) . ν(i0) l! i0=0 l=0 We claim that for some service-level objectives the steady-state probabilities have the same structure as those investigated by Khudyakov [32] and in this work. For example, one can develop the marginal distribution, the probability of delay and blocking, and E[W ].
47 8.1 The marginal distribution
PK We will calculate the marginal distribution π(i, l) as l = j=1 ij. For each (i, l) such as 0 ≤ i+l ≤ n: X π(i, l) = π(i, i1, .., iK )
i1,...,iK |i1+...+iK =l
X 1 i Y 1 ij = π0 (ρa) (ρj) ν(i) ij! i1,...,iK |i1+...+iK =l j∈J 1 1 = π (ρ )i (A)l . 0 ν(i) a l!
8.2 The probability of delay
The second example we show is for the calculation of the probability of delay: X Pn(W > 0) = Pn−1(Q1(∞) ≥ s) = πn−1(i, i1, ..., iK ) s≤i+i +...+i ≤n−1|i≥s 1 K (8.1) X 1 i Y 1 ij = π0 i−s (ρa) (ρj) s!s ij! s≤i+i1+...+iK ≤n−1|i≥s j∈J where,
−1 X 1 i0 Y 1 ij π0 = (ρa) (ρj) ν(i0) ij! 0≤i0+i1+...+iK ≤n−1 j∈J n−1 n−i−1 X X 1 i0 1 l = (ρa) (A) . ν(i0) l! i0=0 l=0 These probabilities have the same structure as that which was investigated in Chapter5. Therefore, we can conclude with the following general theorems:
Theorem 8. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the following conditions:
n − s − A lim √ = η, −∞ < η < ∞, (i) λ→∞ A √ ρ lim s 1 − a = β, −∞ < β < ∞, β 6= 0 (ii) λ→∞ s where all other parameters are fixed. Then
R β √ −1 −∞ Φ η + (β − t) B dΦ(t) lim P (W > 0) = 1 + √ (8.2) λ→∞ φ(β)Φ(η) φ( η2+β2) 1 2 2 η1 β − β e Φ(η1) where η = η − √β . 1 B
48 Theorem 9. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the following conditions: n − s − A lim √ = η, −∞ < η < ∞; (i) λ→∞ A √ ρ lim s 1 − a = β, β = 0 (ii) λ→∞ s where all other parameters are fixed. Then R 0 √ −1 −∞ Φ η − t B dΦ(t) lim P (W > 0) = 1 + (8.3) λ→∞ √1 √1 (ηΦ(η) + φ(η)) B 2π where η = η − √β . 1 B
8.3 The probability of blocking
We can also calculate the probability of blocking. From the stationary probability, we will deduce the probability Pl that there are l customers in the system (0 ≤ l ≤ n). We will use the following relation: l X X X Pl := π(i, i1, ...iK ) = π(i, i1, ..., iK )
i,i1...,iK ≥0 i=0 i1+...+iK =l−i i+i1...+iK =l l K X X 1 i Y 1 ij = π0 (ρa) (ρj) ν(i) ij! i=0 i1+...+iK =l−i j=1 l X 1 1 = π (ρ )i (A)l−i 0 ν(i) a (l − i)! i=0 s n ! X 1 1 X 1 1 = π (ρ )i (A)l−i + (ρ )i (A)l−i 0 i! a (l − i)! s!si−s a (l − i)! i=0 i=s+1 This phrase is exactly the same phrase that was investigated in Chapter5. Thus, we can state the following theorem:
Theorem 10. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the following conditions: n − s − A lim √ = η, −∞ < η < ∞; (i) λ→∞ A √ ρ lim s 1 − a = β, −∞ < β < ∞, β 6= 0 (ii) λ→∞ s where all other parameters are fixed. Then
2 p η1 √ νφ(ν )Φ(ν ) + φ( η2 + β2)e 2 Φ(η ) lim sP (block) = 1 2 √ 1 (8.4) λ→∞ β √ φ(β)Φ(η) φ( η2+β2) 1 2 R 2 η1 −∞ Φ η + (β − t) B dΦ(t) + β − β e Φ(η1)
49 √ √ −1 −1 where η = η − √β ,ν = η√ B +β ,ν = β√ B −η ,ν = √ 1 . 1 B 1 1+B−1 2 1+B−1 1+B−1 Theorem 11. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the following conditions: n − s − A lim √ = η, −∞ < η < ∞; (i) λ→∞ A √ ρ lim s 1 − a = β, β = 0 (ii) λ→∞ s where all other parameters are fixed. Then 1 √ νφ(ν1)Φ(ν2) + √ Φ(η) lim sP (block) = 2π (8.5) 0 √ λ→∞ R Φ η − t B dΦ(t) + √1 √1 (ηΦ(η) + φ(η)) −∞ B 2π where ν = √ η , ν = √ η ,ν = √ 1 . 1 1+B 2 1+B−1 1+B−1
8.4 Approximation of E[W]
The last example stated here is for the approximation of E[W].
Theorem 12. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the following conditions: n − s − A lim √ = η, −∞ < η < ∞; (i) λ→∞ A √ ρ lim s 1 − a = β, −∞ < β < ∞, β 6= 0 (ii) λ→∞ s where all other parameters are fixed. Then √ p 2 2 1 η2 −1 2 −1 √ φ(β)Φ(η) + φ( η + β )e 2 1 Φ(η1) B β − ηβ B − 1 lim sE[W ] = √ λ→∞ β √ φ(β)Φ(η) φ( η2+β2) 1 2 2 R 2 η1 µβ −∞ Φ η + (β − t) B dΦ(t) + β − β e Φ(η1) where η = η − √β . 1 B Theorem 13. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the following conditions: n − s − A lim √ = η, −∞ < η < ∞; (i) λ→∞ A √ ρ lim s 1 − a = β, β = 0 (ii) λ→∞ s where all other parameters are fixed. Then √ 1 B−1 (η2 + 1)Φ(η) + ηφ(η) lim sE[W ] = √ √ √ λ→∞ 2µ R 0 −1 2π −∞ Φ η − t B dΦ(t) + B (ηΦ(η) + φ(η)) where η = η − √β . 1 B
50 9 Fluid limits
The model presented in Section2 is difficult to analyze in non-stationary periods, i.e. when con- sidering fluid and diffusion limits. The complication derives from the blocking characteristic of our model. The fact that there is a limit to the number of beds available, turns the diffusion process into a two-sided reflected process, and creates theoretical difficulties (Pang et al. [44] provide a survey of the mathematical machinery involved in this case). As a first stage, we decided to analyze a simpler model, which was described in general terms in Section 2.3.3. The system was presented in Figure8. It is a state-dependent open queueing network. There are two articles that provide the mathematical framework for finding fluid and diffusion limits of such systems. The first was presented by Mandelbaum and Pats [39] on state-dependent stochastic networks, and the second by Mandelbaum et al. [35] on time-varying queues. As a start, we shall use the method of Mandelbaum and Pats [39]. First, we represent this network in their format.
exp(μ ), S
μμ()QSQ=∧ ( ) 1 Poiss()λ 1-p λ()Q = λ 1
p
exp(δ ),∞
δ ()QQ= δ 2
2
The medical Center: Figure 15: Version 3
K The model is actually a 2-node state-dependent stochastic network denoted as (Mξ/Mξ/Sk) Services MU1 where Sk ∈ {1, 2, ..., ∞}, k ∈ 1, 2, ..., K and K = 2. Let Q = {Q(t), t ≥ 0} be a 2-dimensional stochastic queueing process, where Q(t) = (Q1(t),Q2(t)): Q1(t) representing the number of needy MU2 patients in the system (i.e., those either waiting for service or being served), and Q2(t) the number Patient of dormant patients in the system,Emergency at time t. The process Q satisfies: Arrivals Ward MU3 discharge
Q(t) = Q(0) + A(t) + F (t) − D(t),
Blocked Patient MUn patients discharge 51
where for each k = 1, 2:
Z t + Ak(t) = Nk λk(Q(u))du , 0 K X Z t Fk(t) = 1 {Uj[Sj(u)] ∈ πjk(Q(u−))} dDj(u), j=1 0 Z t Dk(t) = 1 {Qk(u−) > 0} dSk(u), 0 Z t − Sk(t) = Nk µk(Q(u))du . 0
Here Ak(t) is the input stream into the system entering node k, Fk(t) is the input stream into node k from other nodes in the system, Dk(t) is the actual output stream from node k, and Sk(t) ∞ is the potential output stream from node k; {Uj[l]}l=0 are sequences of i.i.d. random variables, + − uniformly distributed on [0, 1]; Nk ,Nk are standard independent Poisson processes. Specifically, in our system:
Z t + A1(t) = N1 λdu , 0
A2(t) = 0, Z t F1(t) = 1dD2(u) = D2(t), 0 Z t F2(t) = 1 {U1 ∈ [0, p]} dD1(u), 0 Z t D1(t) = 1 {Q1(u−) > 0} dS1(u), 0 Z t D2(t) = 1 {Q2(u−) > 0} dS2(u), 0 Z t − S1(t) = N1 µ(S ∧ Q1(u))du , 0 Z t − S2(t) = N2 δQ2(u)du . 0
n n K Now consider a sequence of systems as described above indicated by (Mξ /Mξ /Sk) , n = 1, 2, ..., each of them specified in the same way, with the following changes:
n n λ1 (Q (u)) = nλ,
n λ2 (·) = 0, Sn Qn(u) µn(Qn(u)) = nµ ∧ 1 , 1 n n Qn(u) µn(Qn(u)) = nδ 2 . 2 n
52 Let qn(t) = {qn(t), t ≥ 0}, n ≥ 1, be the two-dimension fluid scaled process. Qn(t) qn(t) ≡ , n ≥ 1. n Assume that, for each k = 1, 2, 1 1 1 λn(nξ) → λ (ξ), µn(nξ) → µ (ξ), Sn → s,¯ u.o.c., n k k n k k n as n ↑ ∞ (plus a few more technical conditions, such as Lipschitz smoothness of the rates). Then by the FLLN in [39] [Theorem 4.6], {qn} converges, u.o.c. (uniformly on compact) over [0, ∞) in probability, as n ↑ ∞, to a deterministic absolutely continuous function q. Indeed, q is the unique solution to the following differential equation:
R t q(t) = q(0) + θ(q(u))du 0 t + R [I − P T (q(u))]dy(u) ≥ 0, t ≥ 0, 0 (9.1) y is nondecreasing in each coordinate, y(0) = 0, R ∞ T 0 1 [q(t) > 0]dy(t) = 0, (this is based on the Martingale representation of the process), where
θ(·) = λ(·) + [P T (·) − I]µ(·), (9.2) and Z · y(·) = I {q(u) = 0} µ(q(u))du. 0 In our system
λ1(q(u)) = λ, λ2(·) = 0,
µ1(q(u)) = µ(¯s ∧ q1(u)),
µ2(q(u)) = δq2(u), 0 p P (·) = . 1 0
Thus, λ −1 1 µ(¯s ∧ q1(u)) θ(·) = + . 0 p −1 δq2(u)
As shown in Mandelbaum and Pats [39], in order to solve (9.1) we should solve the following differential equation:
q˙ (t) = θ (q (t)) + I − P T (q(t)) m˜ (t), (9.3)
53 wherem ˜ (t) =y ˙ (t), with the initial condition q(0). In our case: λ −1 1 µ(¯s ∧ q1(u)) 1 −1 q˙ (t) = + + m˜ (t) (9.4) 0 p −1 δq2(u) −p 1
µ(¯s ∧ q1(t)) m˜ (t) =y ˙(t) = I {q(t) = 0} µ(q(t)) = I {q(t) = 0} , δq2(t) considering the fact that when q(0) ≥ 0, q(t) ≥ 0, for all t ≥ 0, and that in fact q(0) ≥ 0, equation (9.4) is equivalent to the following Differential Equation (DE) system: dq1 (t) = λ + δq2(t) − µ(¯s ∧ q1(t)) dt (9.5) dq2 dt (t) = −δq2(t) + pµ(¯s ∧ q1(t))
In the next subsection we are analyzing this DE in two scenarios: a) steady-state analysis (as t → ∞) – finding if there exists one and what is the system equilibrium, by solving the system of dq equations: dt (t) = 0, and b) transient analysis (for all t) – finding a closed-form solution for that DE system. As mentioned before, one can also use the mathematical framework of Mandelbaum et al. [35]. Our model is actually a special case of a time-varying queue, where the arrival rate λ is actually constant. Using this framework will give us a more general solution that can be used for more general settings, such as time-varying arrivals, and time-varying staffing policies. Note that with time-varying arrival rates (λ(t), t ≥ 0), the DE system is unlikely to be tractable analytically. In the future, we might try to solve it numerically, as was done by Mandelbaum et al. [36, 37, 38].
9.1 Steady-state analysis of the fluid system
By Mandelbaum and Pats [39], the system (9.5) does have a unique solution. We find the system dq equilibrium (t → ∞) by solving dt (t) = 0. In Section 9.2, we show that the limit is finite or infinite, in accordance to certain conditions. When that limits (q1(∞), q2(∞)) is finite, if q1(∞) < s¯ then:
λ + δq2(∞) − µq1(∞) = 0,
− δq2(∞) + pµq1(∞) = 0.
Thus,
λ pλ q (∞) = , q (∞) = . 1 (1 − p)µ 2 (1 − p)δ
54 If q1(∞) ≥ s¯ then:
λ + δq2(∞) − µs¯ = 0
− δq2(∞) + pµs¯ = 0
Thus,
pµs¯ µs¯ − λ q (∞) = Arbitrary, q (∞) = = . 1 2 δ δ
λ This system will have a solution only ifs ¯ = (1−p)µ . In this case q2(∞) is determined uniquely. As pµs¯ for q1(∞); we shall see in the following section that it is given by q1(0) + q2(0) − δ , which leads to the following equlibrium:
pλ pλ q (∞) = q (0) + q (0) − , q (∞) = . 1 1 2 (1 − p)δ 2 (1 − p)δ
9.2 Transient analysis of the fluid system
The DE system (9.5) is a first order piece-wise linear DE system. In order to find a solution to this 2 DE, one must divide the domain (R+ = {(q1, q2), qi ≥ 0}) into ranges, two in our case (range I: q1 < s¯, range II: q1 ≥ s¯). A closed-form solution for this system can be found as follows: as a start, one must find a general solution to each of the linear parts, separately. Then one substitutes the initial conditions of the first range. After this, one combines (or patches) the two ranges by placing the boundary value of the first domain to be the initial condition of the second domain, etc. In order to decide how the function behaves on the boundary (i.e. whether the function reflects back to the first domain, moves to the second domain, or stays on the boundary), one must investigate the DE itself at its non-smooth points. Accordingly, a general solution for (9.5) was found using
MATLAB, for each range. The first is for q1(0) < s¯ : 1 pλ q (t) = C · AeAt + C · BeBt + δ C eAt + C eBt + 1 pµ 2 1 2 1 (1 − p)δ
At Bt pλ q2(t) = C2 · e + C1 · e + (1 − p)δ (9.6) 1 1 1p A = − δ − µ + (δ2 − 2µδ + µ2 + 4pµδ) 2 2 2 1 1 1p B = − δ − µ − (δ2 − 2µδ + µ2 + 4pµδ). 2 2 2
Here C1 and C2 are set by the initial conditions in the following way: C B + C A + δq (0) q (0) = 1 2 2 1 pµ (9.7) pλ q (0) = C + C + 2 1 2 δ(1 − p)
55 thus, pλ C = q (0) − − C 1 2 δ(1 − p) 2 pµq (0) − (δ + B)q (0) pλ B C = 1 2 + 2 A − B δ(1 − p) A − B
For example, if q1(0) = q2(0) = 0 then the constants C1 and C2 are: pλ A pλA C1 = − = − > 0 δ(1 − p) A − B δ(1 − p)pδ2 − 2µδ + µ2 + 4pµδ (9.8) pλ B pλB C2 = = < 0. δ(1 − p) A − B δ(1 − p)pδ2 − 2µδ + µ2 + 4pµδ
When q1(0) ≥ s¯ the solution is:
pµs¯ −δt pµs¯ q1(t) = (λ − (1 − p)µs¯) t − q2(0) − e + q2(0) − + q1(0) δ δ (9.9) pµs¯ pµs¯ q (t) = + q (0) − e−δt, 2 δ 2 δ where q1(0) and q2(0) are the initial conditions. Actually, the existence of a smooth solution to the
DE assures us a smooth-fit of the two ranges, if at some time t0, t0 < ∞, q1(t0) =s ¯, then both equations must be satisfied. Mandelbaum and Pats [39] have assured us the existence and uniqueness of a deterministic limit function; the remaining question is the form of that function. It is easy to see that the system reaches equilibrium a.s. (almost surely), which is a function of the starting point q(0), by checking the convergence of the solutions found in (9.6) and (9.9). The parameters λ, µ, δ, p, ands ¯ are all positive, therefore, one concludes that A and B are negative; since:
1 1 1p 1 1 1p A = − δ − µ + (δ2 − 2µδ + µ2 + 4pµδ) < − δ − µ + (δ2 − 2µδ + µ2 + 4µδ) 2 2 2 2 2 2 1 1 1p = − δ − µ + (δ + µ)2 = 0 2 2 2 1 1 1p B = − δ − µ − (δ2 − 2µδ + µ2 + 4pµδ) < 0 2 2 2 Hence, all the exponents in (9.6) converge to 0 as t tends to infinity. Thus, the limit of (9.6) is given by: λ lim q1(t) = t→∞ (1 − p)µ (9.10) pλ lim q2(t) = . t→∞ (1 − p)δ Notice that the limit value in this range does not depend on the initial values, and that the function q(t) is increasing/decreasing according to C1 and C2. Naturally, this limit is exactly the equilibrium found earlier.
56 In addition, since δ is positive, the exponents in (9.9) also converge to 0 as t tends to infinity. Thus, the limit of (9.9) is given by:
pµs¯ q2(0) − δ + q1(0) if λ − (1 − p)µs¯ = 0, lim q1(t) = ∞ if λ − (1 − p)µs¯ > 0, (9.11) t→∞ q1(t∞) =s, ¯ t∞ < ∞ otherwise.
pµs¯ lim q2(t) = . t→∞ δ In here, the limit value is a function of the initial value. Notice that the system might converge to equilibrium in range II only if λ − (1 − p)µs¯ = 0, which is the exact condition found earlier in λ Section 9.1 (i.e.s ¯ = (1−p)µ ).
Therefore, for investigating the convergence to the limit values q(∞) = (q1(∞), q2(∞)) we con- sider three cases separately. To date, we have not finished analyzing all the cases, we state here the partial results we so far obtained. In general we conjecture the following: if the system is overloaded λ i.e.s ¯ < (1−p)µ , then if q1(0) < s then at some point in time q1(t) > s¯ the function will pass to area λ II and explode, not reaching an equilibrium; if the system is underloaded i.e.s ¯ > (1−p)µ , then the λ function will reach the equilibrium inside range I, and if the system is critically-loaded i.e.s ¯ = (1−p)µ the function may reach equilibrium in the either range I or II. Figure 16 illustrate the convergence of the DE in the three cases described above. The system parameters of this illustration are given in Table6. The red color represents cases that are converging in range I, the blue color represents cases that are converging in range II, and the purple is for cases that do not converge at all (i.e. they explode).
λ δ µ p s – overloaded s – critically-loaded s – underloaded 10 0.5 1 0.5 10 20 30
Table 6: Parameters for illustration of the fluid DE convergence
57 (critically-loaded)
(overloaded) (underloaded)
Figure 16: Fluid DE convergence
58 λ Case 1: underloaded system. Heres ¯ > (1−p)µ . If q1(0) > s¯ then Equation (9.9) is the relevant one, and λ−(1−p)µs¯ < 0, which means that at some point in time q1(t) will start decreasing monotonically (almost linearly) without converging, till it will reach the values ¯ (according to (9.11)) and pass to the first range and converge to (9.10). Therefore, if q1(0) ≤ s¯ from the same reasons, we expect q(t) to change till it will converge in range I to the equilibrium (9.10). Note that in this case q1(∞) < s¯, which means that both the number of customers (patients) in the queue and the probability to wait tend to 0. λ Case 2: critically-loaded system. Heres ¯ = (1−p)µ . In the case where q1(0) < s¯, we have only analyzed the specific case of q(0) = (0, 0). Assuming q(0) = (0, 0), as long as q1(t) < s¯, Equation
(9.6) is the relevant one. Note by (9.8), that in this case C1 > 0 and C2 < 0. The convergence pλ here is based on the fact that q2(t) is monotonic always less than (1−p)δ . This is proved using the dq2(t) At Bt following analysis: we need to prove that dt = C2Ae + C1Be > 0. If q(0) = (0, 0), then according to (9.7) C2A + C1B = 0. Thus,
At Bt At Bt Bt At C2Ae + C1Be = −C1Be + C1Be = C1B(e − e ).
Bt At dq2(t) At We know that C1 > 0, B < 0 and that e − e < 0 since |A| < |B|. Therefore, dt = C2Ae + Bt C1Be > 0 for all t, and q2(t) is strictly increasing. Since the exponents converge to 0, q2(t) ≤ pλ (1−p)δ .
Now, we need to investigate q1(t). In our DE, the pasting point is q1 =s ¯. At this point, we want to know whether the function reflects back to the first domain (i.e. q1 decreases), moves to the second domain (i.e. q1 increases), or stays on the edge (i.e. q1 =s ¯). In order to know that we must
dq1(t) find conditions in which dt = 0. According to (9.5), this happens when λ+δq2(t)−µs¯ = 0, which µs¯−λ µs¯−λ dq1(t) is equivalent to q2(t) = δ . Thus, if at the point of reaching the edge q2(t) < δ then dt < 0, µs¯−λ and q1 will decrease, and the solution of the first domain will be the relevant one; if q2(t) > δ dq1(t) then dt > 0, and q1 will increase, and the solution of the second domain will be the relevant one. µs¯−λ dq1(t) dq2(t) If both q2(t) = δ and dt = dt = 0, then the system reaches equilibrium. pλ µs¯−λ In our case q2(t) < (1−p)δ = δ for all finite t, which means that if q1(t) =s ¯ for some finite t, the function reflects to area I, thus, when t → ∞ the convergence is reached in area I, to the equilibrium (9.10).
If q1(0) ≥ s¯ then Equation (9.9) is the relevant solution of the DE. In this case, q1(t) converges pµs¯ to a constant q1(∞) = q2(0) − δ + q1(0), which is determined by the starting conditions, and pµs¯ pλ pµs¯ q2(t) converges to q2(∞) = δ = (1−p)δ . In here, the function q(t) is monotonic; if q2(0) > δ pµs¯ then q1(t) is strictly increasing, always more than s and less than q2(0) − δ + q1(0), and q2(t)
59 pµs¯ is strictly decreasing always more than δ ; they jointly converge to the equilibrium noted earlier. pµs¯ pµs¯ If q2(0) < δ then exactly the opposite occurs; q2(t) is strictly increasing always less than δ , pµs¯ and q1(t) is strictly decreasing always more than q2(0) − δ + q1(0). Since q1(t) is decreasing, we want to define a threshold value from which the convergence is taking place within range I. Since the pµs¯ pµs¯ equilibrium is q2(0)− δ +q1(0), the threshold is determined by the equation: q2(0)− δ +q1(0) =s ¯, pµs¯ which means that if q1(0) ≥ s¯−q2(0)+ δ then q1(t) > s for all t and it converges to the equilibrium pµs¯ noted above, but if q1(0) < s¯ − q2(0) + δ , then at some time t0 q1(t0) =s ¯ and the function move pµs¯ pλ pµs¯ to the first area. Finally, if q2(0) = δ = (1−p)δ , then q2(t) = δ and q1(t) = q1(0) for all t, and the function is at equilibrium. pλ Note that for the special case in which q(0) = (¯s, (1−p)δ ) then also by the DE itself the differential pλ is 0 and thus, q(t) = (¯s, (1−p)δ ) for all t ≥ 0. λ Case 3: overloaded system. Heres ¯ < (1−p)µ . One can easily observe that if q1(0) < s¯ then for small t, t < t1, Equation (9.6) is the relevant one, by which q1(t) is approaching the quantity λ (1−p)µ , therefore, at some point t1, q1(t0) will equals ¯ (i.e. t1 = inf{t : p1(t) =s ¯}) and by the continuity of the solution [39], from that point we can observe its behavior by the second range. If q1(0) > s¯, then by Equation (9.11) q1(t) explodes. Thus, in any starting point the system explodes.
60 10 Defining optimal design
The goal of this chapter is to outline very roughly some methods for optimizing the system using the approximations calculated in previous chapters. This optimization could be done in various ways. For example, from an economic point of view one can find the optimal number of beds and nurses, so that the total cost is minimized while, at the same time, maintaining a predefined service level. The constraints at the service level could be on the delay probability P (W > 0) < a, on the probability of waiting more than t units of time P (W > t) < b, or on the probability of blocking P (block) < c. Any combination of these measures could be used as well.
Define Cn to be the annual bed costs due to space and maintenance, and Cs the annual nurse (servers) costs. Then our optimization problem can be:
minn,s C(n, s) = Cnn + Css
s.t P (W > t) < b
P (block) < c
0 ≤ s ≤ n.
Another possibility could be to look at the situation as a revenue maximization problem. The hospital charges the insurance companies for each patient being hospitalized (under supervision on the necessity of the procedure). A patient that is being blocked is a lost customer to the system as well as a threat to the good reputation of the hospital. Define R to be the annual revenue due to bed occupancy, and OC(n, s) the average bed occupancy level in a system with n beds and s nurses. This type of optimization problem can be formalized as:
maxn,s R(n, s) = R · OC(n, s) − Cnn − Css
s.t P (W > t) < b
P (block) < c
0 ≤ s ≤ n.
One can also solve process-based optimization (control) problems, in support of real-time man- agement, and these will be contemplated on in due time.
61 11 Further research
11.1 Near Future
First, we intend to finish the approximation proofs, design and perform an experiment for analyzing the cases where the approximations work best, and answer questions such as from which system size (i.e. what n) we can use our approximations. The next stage will be validation of the model and the management policies we developed. Validation of theoretical models via collecting, analyzing and experimenting with real data is a prevalent approach in basic sciences (Physics, Biology). Moreover, Call Center models have been validated using data from a large banking system [49]. We intend to validate our models against real-data, refining them if necessary and possible. We hope that during the period of the proposed research, the Technion-affiliated hospital will install a patient-tracking system, based on Radio-Frequency IDs (RFID); see [33] for general RFID Health-Care applications, and [27] for a specific example that uses RFID data in analyzing patient flow in an outpatient clinic. If so, this will provide us with patient-flow data in hospitals, that give accurate knowledge on the processes that patients undergo during their stay. For a start, this data can help us analyzing patients’ flow statistics, and medical staff service time distributions. We also wish to use the RFID data to identify situations where the Health-Care system already operates in the QED regime, and to examine the suitability of these situations for our model recommendations. Verifying our model is possible also by using simulation tools [41]. Using simulation we can examine the fitness of our model to close-to-reality situations. At the same time, we would like to try and develop diffusion models, as a natural extension of the fluid models we started developing in Chapter9. We hope to also try to develop fluid and diffusion models for the finite model, using the framework of Pang et al. [44]. In the next stage, we might continue this research, by extending our model in various ways. The model we used has natural extensions, such as multi-classes of patients or nurses, additional phases of clinical treatment, adding doctors (most likely working in the ED regime, in parallel to nurses in the QED regime), time-variability and random parameters. Hopefully, once we obtain appropriate hospital data, we can use them to help determine those extensions, if any, which yield the best model validity. We have given some thought to some possible extensions as mentioned above, and we now describe some of them, in greater detail, in the following subsections.
62 11.2 Combining managerial / psychological / informational diseconomies-of- scale effects
In their article Boudreau et al. [7] discussed the importance of combining Operation Management and Human Resource Management (HRM) models. The integration between the two fields is very challenging. This integration requires some collaboration with researchers from the field of HRM, which is not common practice. We consider it important to try and carry out such integration in nurse staffing for one main reason; when discussing our models with doctors from an Israely Medial Center, there was major concern that the model might recommend “too large” departments. The claim was that there is a managerial / psychological / informational limit to the number of patients one MU can actually treat, and if that limit is surpassed, the quality-of-care deteriorates. The claim was that a large MU is inferior to several small MUs, even though in a smaller MU one has fewer medical personnel. We found that claim interesting, and suggest a few explanations for the source of this diseconomies-of-scale effect.
1. Managerial causes. The MU manager is the one most responsible for the medical decisions. The doctors work as a team, consulting one another about the patients before taking a medical decision. Adding more doctors thus increases the level of knowledge of the MU but, due to the form of responsibility distribution, it might not always lead to a similar rise in capacity. The framework of team vs. individual responsibility has been investigated in the field of HRM. In our context, one can find the two ways of setting, i.e. (a) team vs. (b) individual. (a) A team of doctors was mentioned above. One can say that nurses also work as a team if all of them are jointly responsible for all the patients in the ward. (b) In the individual setting, there is one nurse for one or more specific patient(s) as in the case of Intensive Care Units. From the operational point of view, the two settings are different.
2. Psychological causes. Are there unique learning and forgetting effects? It seems that there is a difference in learning schemes between various service environments, such as the machine- repairman problem, call center and nursing staffing problems. We know that in call centers, each customer is different, though his/her problems are alike (and can be classified into certain types). One can see a learning curve in which as a new service person starts to work she begins to learn the different types of services; as she deals with more customers she becomes more professional which, in turn, is reflected in the rise in the quality and the efficiency of the given services. This learning effect is known also in Industrial Engineering. This means that, in reality, the service rate µ is not a constant parameter but is actually a function of time,
63 i.e. µ(t). Usually we ignore this effect by looking at steady state, assuming that all nurses have sufficient experience. In repair-man and nurse-staffing problems we see the same effect: as one service person deals with more customers (i.e. machines or patients) she learns more, becoming an expert. An expert will deal with problems more efficiently; she is capable of treating more customers better and faster. But there is one difference: in nursing and machine- repair problems each customer “calls” the server several times during his stay/use, but the server needs to treat each customer as an individual and to remember his problems. This is customer learning effect. Thus, as the number of patients per nurse increases we might find a dis-economic-of-scale effect, where the reset-time of starting the treatment of a patient might grow with the number of customers. One could say that µ is actually a function of the system, for example: µ(s, n). One might be able to make some assumptions on the shape of that function (e.g. convex or constant). Combining the two learning effects, the customer learning and the job learning, one may define a service function µ(s, n, t). But as mentioned before, we may prefer to look at steady state, and assume that all nurses have sufficient experience, and ignore the time effect. The shape of the function µ(s, n) is not clear; it might increase due to overload effects (see below) or decrease as usually happens in learning curves.
3. Informational causes. How does the medical personnel react to the information overload caused by a large number of patient? Hall and Walton [24] reviewed some literature on Information overload in Health-Care systems, which raises some possible effects of overload. This raises the following question: Does the number of errors rise as a function of the nurse-to-patient ratio or as a function of the unit size?
These possible explanations could be investigated; with the combination of the Mental- and Physical-capacity into a single model is being important and challenging. Technically speaking, if we could define some Quality-of-Service (QoS) function in one of the following ways: (a) As a function of the workload (b) As a function of the number of patients in the system, i.e. f(n), or (c) As a function of the number of servers and patients in the system, i.e. f(s, n), then we could combine this QoS function into our optimization model. The following ways are possible:
64 Define E(QoS) - the average Quality of Service in the MU with n beds and s nurses, or define P (QoS < α) - the probability that the service quality in the MU will be less than α. Then,
minn,s C(n, s) = Cnn + Css;
s.t P (W > t) < a;
P (block) < b;
P (QoS < α) < c; or E(QoS) < c
0 ≤ s ≤ n;
The calculation of the QoS measure will be based on the system product-form solution, or its approximations.
11.3 Phases of treatment or Heterogeneous patients
11.3.1 Combining the phases of treatment during hospitalization period
When discussing this research with some medical crew, we became aware of an interesting phe- nomenon; at the beginning of the hospitalization period (approximately the first twenty-four hours), the patient requires intensive care while, as days pass, the care becomes less and less intensive. One can model this fact as a frequency reduction or as a service-duration reduction over time (i.e. the service-time function will decrease as a function of the LoS). We can divide the stay into a finite number of sequences, and categorize them. I will demonstrate this with the two following classes: (A) Intensive Care and (B) Regular Care. The following Figures, 17 and 18, illustrate two possible models of the system. The first assumes that a patient can move from class A to B and the reverse. The second model assumes that a patient starts in class A and, at some point in time, he moves to class B and later leaves the system from class B. In both models the differentiation between the stages was modeled through frequency reduction. This was achieved by using the following assumption: δ > γ. The first system is a Jackson network, with the same structure as described in Chapter8. Thus, it can be solved and approximated using the same methods. (open questions: is there an assumption on the relationship between the sorting probabilities? how can we calculate them?) The second suggestion is a closed BCMP network3 (which has a product-form solution defined by Baskett et al.
3BCMP network contains an arbitrary but finite number N of service centers, and an arbitrary but finite number R of different classes of customers. Customers travel through the network and change class according to transition probabilities. Thus, a customer of class r who completes service at service center i will next require service at center j
65 Jennings (2007):
exp(γ ),n , γ →∞ exp(μ ), s n γγ()QI= I n {}{}nQ−−>12 Q00 QE > μμ()QsQ=∧ ( ) 1 Poiss()λ 1-p E 1 λ n = λn p
exp(δ ),∞ exp(θ ) δ n ()QQ= δ θ n ()QQ= θ 2 E 2
Jennings (2007): Phases of treatment: exp(γ ),n , γ →∞ exp(μ ), s n exp(μ ), S γγ()QI= I n {}{}nQ−−>12 Q00 QE > μμ()QsQ=∧ (r (r=1-p-q)1 ) Poiss()λ Poiss()λ 1 1-p E 1 n λ = λn p exp(δ ),∞
exp(p δ ),∞ exp(θ ) A n n δ ()QQ= δ 2 θ ()QQ= θ E q N 2 exp(γ ),∞
Phases of treatment: B
exp(μ ), S
r (r=1-p-q) Poiss()λ 1
exp(μ ), S Figure 17: Phases of hospitalization - Model 1 exp(δ ),∞ A,()Poiss λ p A
q N exp(γ ),∞
exp(δ ),∞ N B
A
exp(μ ), S exp(γ ),∞
A,()Poiss λ B
exp(δ ),∞ N A
exp(γ ),∞
B
Figure 18: Phases of hospitalization - Model 2
66 [5]), assuming FCFS discipline with identical and exponential service rates. This model describes the situation more precisely, but it might need to be solved separately. Note: This is one of the differences between the call center or repairman problem and nurse staffing. Is it similar to a learning effect in some sense? (the learning of specific customer instead of the learning of the server)
11.3.2 The influence of time delays before and after medical analysis or surgery
In addition to the description of classes of patients in the previous subsection, one can add classes regarding the treatment stages, i.e. before and after medical analysis or surgery. The model shown in Figure 18 also fits this situation. This modeling allows us to give priority in medical care to each class of patients, by setting different waiting threshold for each class. The questions that arise here are about priority schemes, and staffing levels. One might also want to look at a more delicate issue, and check with doctors the possible influence of time delays on the patient situation. Is there a change in the rate of patient’s state during the waiting time, as a function of the patient’s class? Our assumption throughout this work was that this rate is constant, i.e. there is a linear connection between the waiting and the patient’s state. But if it is not, for example, if it is an increasing function, then the model might give an underestimation for the required staffing levels. (Remark: in the case of the machine-repairman problem it could be either way, depending on the mechanical aspects of the machine; there are “delicate” machines that need repairing immediately, while other machines can stay for weeks without changing their state). This function could also change over stages, which means that when a patient’s state is critical, the rate might be increasing, but when the medical state of the patient is stabilized (i.e. not critical) the rate might be constant. (For a model that combines the changing influences of waiting and service (in the psychological aspects) see Carmon et al. [9]).
11.3.3 Classes of patients (Heterogeneous patients)
There are situations where the MU treats a few types of patient, who could be divided into several groups (or classes) that are independent of each other. There are situations where the classes of in class s with a certain probability denoted Pi,r,j,s. There are four types of BCMP networks, that differ according to several condition it satisfies concerning the service discipline and service-time distribution. We will consider here only Type 1. The conditions are: The service discipline is FCFS; all customers have the same service time distribution at this service center, and the service time distribution is a negative exponential. The service rate can be state-dependent where µ(j) will denote the service rate with j customers at the center.
67 patients are dependent in various ways; we dealt with such models in the sections above. When the classes are independent, the model is simpler. An example of a two-class model, can be viewed in Figure 19. The network is a simple BCMP network, where patients from one class cannot transfer to another class but stay in the same class for their complete stay in the system.
exp(μ ), S
Poiss()λ
λ =+λλA B
exp(δ ),∞ N A
exp(γ ),∞
B
Nurses andFigure Doctors: 19: New model for two classes of patients
exp(μ ), S 11.4 Nurses in the QED regime, and doctors11 in the ED regime
N There is a difference between the cost of doctors and of nurses. This difference suggests that a q different regime must be considered when settingexp( staffingμ22 ), S levels for each type. The more expensive p resource should be ED-staffed, gaining very high utilizationD levels, while the less expensive resource should be QED-staffed, with a balance between utilization and service level. We can model this exp(δ ),∞ situation in the following way: we assume that nurses and doctors are separately required, therefore, N 1 the communication and synchronization between nurses and doctors are made at separate times, i.e. not in front of the patient. The model contains two stations of the M/M/S type with sN and sD servers (denote as node N for Nurses, and node D for Doctors), and one M/M/∞ node, as seen in Figure 20. In this way it is a closed network, with a fixed population of n patients. The situation could have been modeled also as a semi-open network that contains one entrance node of the type M/M/1, as done with the model we analyzed previously (in Chapter2). In this case we belive, based on our previous experience and knowledge of the ED and QED regimes, that the appropriate
68
exp(μ ), S
Poiss()λ
λ =+λλA B
exp(δ ),∞ N A
QED+ED conditions might be: exp(γ ),∞ n − sN − sD − ρ1 lim √ = η, B −∞ < η < ∞; (i) λ→∞ ρ1 √ RN lim sN 1 − = β1, −∞ < β1 < ∞; (ii) λ→∞ s1 RD lim sD 1 − = β2, −∞ < β2 < ∞. (iii) Nursesλ→∞ and Doctors:s2
exp(μ11 ), S
N
q exp(μ22 ), S p D
exp(δ ),∞
N 1
Figure 20: ED (doctors) and QED (nurses) model
Another option is to assume that, in some cases, nurses and doctors need to provide service together. This assumption might lead to a different model.
11.5 The combination of patient-call treatments and nurse-initiated treatments
In reality, patients that do not “call” a nurse need to be checked and treated every fixed period of time. This means that if the patient has not called the nurse for T units of time, he will join the queue anyway, since the nurse will initiate the treatment. The consequence is that the time between inter-arrivals is not exponential but truncated-exponential, and the nurse station is of type G/M/s. There are other modeling possibilities, such as using a multi-class chain in which one can divide the flow out of the nurse node into two classes, each with different exponential dormant parameters, as was done in the case of heterogeneous patients above (see Section 11.3).
69 11.6 Integrating IWs with the EW as in Jennings and V´ericourt
Jennings and V´ericourt(2007) recently presented a new possible model for integrating the EW and the IW. It is presented here, as we understood it, in Figure 21. The main differences between their approach and ours are: (a) they assumed abandonments instead of blocking. (b) the approximation Jennings (2007): method they used was different.
exp(γ ),n , γ →∞ exp(μ ), s n γγ()QI= I n {}{}nQ−−>12 Q00 QE > μμ()QsQ=∧ ( ) 1 Poiss()λ 1-p E 1 λ n = λn p
exp(δ ),∞ exp(θ ) δ n ()QQ= δ θ n ()QQ= θ 2 E 2
Phases of treatment: Figure 21: New model for nurse staffing and bed allocation according to Jennings and V´ericourt exp(μ ), S (2007) r (r=1-p-q) Poiss()λ 1
11.7Two service stations exp(δ ),∞
p Suppose one has two stations of the M/M/S typeA with s and s servers (denoted nodes 1 and 2 1 2 q respectively), one entrance nodeN of the type M/M/1, and all other nodes are M/M/∞ nodes. We exp(γ ),∞ believe that the appropriate QED conditions will be: B n − s − s − A lim 1√ 2 = η, −∞ < η < ∞; (i) λ→∞ A √ ρ1 lim s1 1 − = β1, exp(μ ),−∞S < β1 < ∞; (ii) λ→∞ s1 √ A,()Poissρλ2 lim s2 1 − = β2, −∞ < β2 < ∞. (iii) λ→∞ s2
But there are also other possibilities, such as working with one node in the QED regime, and the other node in the ED regime (see section 11.4exp(). δ ),∞ N A
exp(γ ),∞
B
70
A Appendix A
We will rewrite Expression 2.1 of π0 . Let l be the number of occupied beds (with patients or in cleaning) and m be the number of patients, where i of them are in the needy state (0 ≤ i ≤ m ≤ l ≤ n). Thus, l = i + j + k and m = i + j. Then: i m−i l−m −1 Pn Pl Pm 1 λ 1 pλ 1 λ π0 = l=0 m=0 i=0 ν(i) (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Ps Pl Pm 1 λ 1 pλ 1 λ = l=0 m=0 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pn Ps Pm 1 λ 1 pλ 1 λ + l=s+1 ( m=0 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pl Ps 1 λ 1 pλ 1 λ + m=s+1 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pm 1 λ 1 pλ 1 λ + i=s+1 s!si−s (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Ps Pl Pm 1 λ 1 pλ 1 λ = l=0 m=0 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pn Pl Pm 1 λ 1 pλ 1 λ + l=s+1 m=0 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pl Pm 1 λ 1 pλ 1 λ − m=s+1 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pl Ps 1 λ 1 pλ 1 λ + m=s+1 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pm 1 λ 1 pλ 1 λ + i=s+1 s!si−s (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Ps Pl Pm 1 λ 1 pλ 1 λ = l=0 m=0 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pn Pl Pm 1 λ 1 pλ 1 λ + l=s+1 m=0 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pl Pm 1 λ 1 pλ 1 λ − m=s+1 i=s+1 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pl Pm 1 λ 1 pλ 1 λ + m=s+1 i=s+1 s!si−s (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Ps Pl Pm 1 λ 1 pλ 1 λ = l=0 m=0 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pn Pl Pm 1 λ 1 pλ 1 λ + l=s+1 m=0 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pn Pl Pm 1 1 λ 1 pλ 1 λ + l=s+1 m=s+1 i=s+1 s!si−s − i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pn Pl Pm 1 λ 1 pλ 1 λ = l=0 m=0 i=0 i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ i m−i l−m Pn Pl Pm 1 1 λ 1 pλ 1 λ + l=s+1 m=s+1 i=s+1 s!si−s − i! (1−p)µ (m−i)! (1−p)δ (l−m)! γ By using the multinomial theorem for the first sum yields:
n l X 1 λ pλ λ π−1 = + + 0 l! (1 − p)µ (1 − p)δ γ l=0 n l m i X X X 1 1 1 λ + − · s!si−s i! (m − i)!(l − m)! (1 − p)µ l=s+1 m=s+1 i=s+1 pλ m−i λl−m · (1 − p)δ γ
71 B Four auxiliary lemmas
In this section we will prove four lemmas that will help us in the proofs of our approximations.
B.1 Proof of Lemma1
Lemma 1. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED conditions.
Define ζ1 as the expression
n−s−1 e−RN 1 X 1 ζ = (R )s (R + R )l e−(RD+RC ). 1 s! N 1 − ρ l! D C l=0 Then φ(β)Φ(η) lim ζ1 = . λ→∞ β √ s s Proof. By using Stirling’s formula s! ≈ 2πs e , and assumption QED (ii), one obtains for ζ1:
s− λ s √ n−s−1 l e (1−p)µ λ s 1 pλ λ pλ λ X − (1−p)δ + γ ζ1 ≈ √ + e 2πs (1 − p)sµ β l! (1 − p)δ γ l=0 λ n−s−1 s− (1−p)µ s l pλ e λ X 1 pλ λ − + λ = √ + e (1−p)δ γ 2πβ (1 − p)sµ l! (1 − p)δ γ l=0 s(1−ρ) e s = √ ρ P (Xλ ≤ n − s − 1) 2πβ
λ where ρ = (1−p)sµ , and Xλ is a random variable with the Poisson distribution with parameter pλ λ RD + RC (where RD = (1−p)δ , RC = γ ). When λ → ∞, RD + RC → ∞ too, since p,δ, and γ are fixed. Note that Xλ − RD − RC n − s − 1 − RD − RC P (Xλ ≤ n − s − 1) = P √ ≤ √ RD + RC RD + RC Thus, when λ → ∞, by the Central Limit Theorem (Normal approximation to Poisson) we have X − R − R λ√ D C ⇒ N(0, 1) RD + RC and due to assumption QED (i) of the lemma we get 4
P (Xλ ≤ n − s − 1) → P (N(0, 1) ≤ η) = Φ(η), asλ → ∞ (B.1)
4Here we use the following theorem (from [7]):
Theorem 14. Let ςn ⇒ ς and Fς - the distribution function of ς is everywhere continuous. Let also
xn → x∞ as n → ∞, where {xn} is a sequence of scalars. Here x∞ ∈ [−∞, ∞]. Then Fςn (xn) → Fς (x∞).
72 where N(0, 1) is a standard normal random variable with distribution function Φ(·). It follows thus that
s(1−ρ) s(1−ρ+ln ρ) e s e ζ1 ≈ √ ρ Φ(η) = √ Φ(η). 2πβ 2πβ
Making use of the expansion
(1 − ρ)2 ln ρ = ln(1 − (1 − ρ)) = −(1 − ρ) − + o(1 − ρ)2, (ρ → 1) 2 one obtains
2 2 s(1−ρ−(1−ρ)− (1−ρ) ) − s(1−ρ) e 2 e 2 ζ1 ≈ √ Φ(η) = √ Φ(η) 2πβ 2πβ by assumption QED (ii) s(1 − ρ)2 → β2, when λ → ∞. This implies
φ(β)Φ(η) lim ζ1 = λ→∞ β where φ(·) is the standard normal density function, and Φ(·) is the standard normal distribution function. This proves Lemma1.
B.2 Proof of Lemma2
Lemma 2. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED conditions.
Define ζ2 as the expression
n−s−1 −(RN +RD+RC ) n−s l e ρ X 1 RD RC ζ = (R )s + . 2 s! N 1 − ρ l! ρ ρ l=0 Then
p 2 2 φ( η + β ) 1 η2 lim ζ2 = e 2 1 Φ(η1). λ→∞ β
Proof. Again according to Stirling’s formula, and assumption QED (ii), one obtains for ζ2:
s− λ − pλ − λ s n−s−1 l e (1−p)µ (1−p)δ γ λ ρn−s X 1 pλ λ ζ2 ≈ √ + 2πs (1 − p)sµ 1 − ρ l! (1 − p)δρ γρ l=0 pλ λ √ n−s−1 s(1−ρ)− (1−p)δ − γ n l pλ e sρ pλ + λ X 1 pλ λ − + λ = √ e (1−p)δρ γρ + e (1−p)δρ γρ 2πs β l! (1 − p)δρ γρ l=0 pλ λ 1−ρ s(1−ρ)+ (1−p)δ + γ ρ e n = √ ρ P (Yλ ≤ n − s − 1) 2πβ
73 λ RD+RC where ρ = (1−p)sµ , and Yλ is a random variable with the Poisson distribution with parameter ρ pλ λ (where RD = (1−p)δ , RC = γ ). Note that RD+RC RD+RC Yλ − ρ n − s − 1 − ρ P (Yλ ≤ n − s − 1) = P q ≤ q RD+RC RD+RC ρ ρ
Now we need to find the limit for the following fraction
RD+RC n − s − ρ q RD+RC ρ as λ → ∞ using assumption QED (i). √ RD+RC RD+RC n − s − ρ η RD + RC + RD + RC − ρ lim q = lim q λ→∞ RD+RC λ→∞ RD+RC ρ ρ √ s √ RD + RC (ρ − 1) sµ(pγ + (1 − p)δ) = lim η ρ + √ = η − lim (1 − ρ) λ→∞ ρ λ→∞ δγ s µ(pγ + (1 − p)δ) = η − β. δγ
Denote s µ(pγ + (1 − p)δ) η = η − β 1 δγ
Thus, when λ → ∞, by the Central Limit Theorem (Normal approximation to Poisson) we have RD−RC Yλ − ρ q ⇒ N(0, 1) RD+RC ρ and
P (Yλ ≤ n − s − 1) → P (N(0, 1) ≤ η1) = Φ(η1), as λ → ∞ where N(0, 1) is a standard normal random variable with distribution function Φ. It follows thus that
pλ λ 1−ρ pλ λ 1−ρ s(1−ρ)+ (1−p)δ + γ ρ s(1−ρ)+ (1−p)δ + γ ρ +n ln ρ e n e ζ2 ≈ √ ρ Φ(η1) = √ Φ(η1). 2πβ 2πβ
Making use of the expansion
(1 − ρ)2 ln ρ = ln(1 − (1 − ρ)) = −(1 − ρ) − + o(1 − ρ)2, (ρ → 1) 2
74 λ q λ and using our assumptions that as λ → ∞: ρ → 1, and s ≈ (1−p)µ + β (1−p)µ , n − s ≈ √ η RD + RC + RD + RC , one obtains pλ λ 1 − ρ s(1 − ρ) + + + n ln ρ (1 − p)δ γ ρ pλ λ 1 − ρ (1 − ρ)2 = s(1 − ρ) + + − n 1 − ρ + (1 − p)δ γ ρ 2 pλ λ n(1 − ρ)2 = − n − s − − (1 − ρ) − (1 − p)δρ γρ 2 2 p RD + RC n(1 − ρ) ≈ − η R + R + R + R − (1 − ρ) − D C D C ρ 2 RD + RC n p = − (1 − ρ)2 − η R + R (1 − ρ) ρ 2 D C √ λ q λ η RD + RC + RD + RC + + β RD + RC (1−p)µ (1−p)µ p ≈ − (1 − ρ)2 − η R + R (1 − ρ) ρ 2 D C
λ ! s R + R RD + RC + 1 λ = D C − (1−p)µ (1 − ρ)2 − β (1 − ρ)2 ρ 2 2 (1 − p)µ √ η RD + RC p − (1 − ρ)2 − η R + R (1 − ρ) 2 D C λ ! s R + R RD + RC + β2(1 − p)µ 1 λ β2(1 − p)µ = D C − (1−p)µ − β ρ 2 λ 2 (1 − p)µ λ
√ 2 r η RD + RC β (1 − p)µ p (1 − p)µ − − η R + R β 2 λ D C λ p 1 p 1 1 ! + γ + γ + ≈ (1−p)δ − (1−p)δ (1−p)µ (β2(1 − p)µ) ρ 2 s s p 1 1 p 1 p 1 − η + βp(1 − p)µ ≈ β2 + (1 − p)µ − 1 − ηβ + p(1 − p)µ (1 − p)δ γ 2 (1 − p)δ γ (1 − p)δ γ s ! 1 1 p 1 = − (η2 + β2) + η2 − 2 ηβ + p(1 − p)µ 2 2 (1 − p)δ γ s !2 p 1 + β + p(1 − p)µ (1 − p)δ γ s !2 1 1 p 1 = − (η2 + β2) + η − β + p(1 − p)µ 2 2 (1 − p)δ γ 1 1 = − (η2 + β2) + η2. 2 2 1
75 Therefore,
pλ λ 1−ρ s(1−ρ)+ + +n ln ρ − 1 (η2+β2)+ 1 η2 e (1−p)δ γ ρ e 2 2 1 lim ζ2 ≈ √ Φ(η1) ≈ √ Φ(η1) λ→∞ 2πβ 2πβ p 2 2 φ( η + β ) 1 η2 = e 2 1 Φ(η ). β 1 This proves Lemma2.
B.3 Proof of Lemma3
Lemma 3. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED or QED0 conditions. Define ξ as the expression
X 1 ξ = (R )i (R )j (R )k e−(RN +RD+RC ). i!j!k! N D C i,j,k|i≤s, i+j+k≤n−1
Then s ! Z β δγ lim ξ = Φ η + (β − t) dΦ(t). λ→∞ −∞ µ(pγ + (1 − p)δ)
Proof. We will find the asymptotic behavior of ξ by finding its lower and upper bounds. Let us l consider a partition {sh}h=0 of the interval [0, s].
sh = s − hτ, h = 0, 1, ..., `; s`+1 = 0
h q λ i where τ = (1−p)µ , is an arbitrary non-negative real and ` is a positive integer. s If λ and s tend to infinity and satisfy the QED assumption (4.2) part (ii), then ` < τ for λ big enough and all the sh belong to [0, s]; h = 0, 1, ..., `. Emphasize that the length τ of every interval [sh−1, sh] depends on λ. The variable ξ is given by the formula (5.3). Let us consider a lower estimate for ξ given by the following sum:
` sh i X X 1 λ − λ ξ ≥ ξ = e (1−p)µ · 1 i! (1 − p)µ h=0 i=sh+1 n−sh−1 j n−sh−j−1 k X 1 pλ − pλ X 1 λ − λ · e (1−p)δ e γ j! (1 − p)δ k! γ j=0 k=0 (B.2) ` sh i X X 1 λ − λ = e (1−p)µ P (Y ≤ n − s − 1) i! (1 − p)µ n h h=0 i=sh+1 ` X = P (sh+1 ≤ Xn ≤ sh)P (Yn ≤ n − sh − 1) h=0
76 λ pλ λ where Xn and Yn are independent Poisson random variables with parameters (1−p)µ and (1−p)δ + γ , respectively. λ If λ → ∞ then (1−p)µ → ∞, since p and µ are fixed. Note that
λ λ λ sh+1 − (1−p)µ Xn − (1−p)µ sh − (1−p)µ P (sh+1 ≤ Xn ≤ sh) = P ≤ ≤ . q λ q λ q λ (1−p)µ (1−p)µ (1−p)µ Thus, when λ → ∞, by the Central Limit Theorem (Normal approximation to Poisson) we have
λ Xn − (1−p)µ ⇒ N(0, 1). q λ (1−p)µ Since q λ λ λ sh − s − h (1−p)µ − (1−p)µ lim (1−p)µ = lim λ→∞ q λ λ→∞ q λ (1−p)µ (1−p)µ λ q λ q λ λ (1−p)µ − β (1−p)µ − h (1−p)µ − (1−p)µ = lim = β − h λ→∞ q λ (1−p)µ we obtain:
P (sh+1 ≤ Xn ≤ sh) = Φ(β − h) − Φ(β − (h + 1)), h = 0, .., ` − 1 (B.3) P (0 ≤ Xn ≤ s`) = Φ(β − `).
pλ λ Similarly, if λ → ∞ then (1−p)δ + γ → ∞, since p,δ and γ are fixed. Note that
pλ λ pλ λ Yn − (1−p)δ + γ n − sh − (1−p)δ + γ P (Yn ≤ n − sh) = P ≤ . q pλ λ q pλ λ (1−p)δ + γ (1−p)δ + γ Thus, when λ → ∞, by the Central Limit Theorem (Normal approximation to Poisson) we have
pλ λ Yn − (1−p)δ + γ ⇒ N(0, 1). q pλ λ (1−p)δ + γ Since
pλ λ q λ pλ λ n − sh − (1−p)δ + γ n − s − h (1−p)µ − (1−p)δ + γ lim = lim λ→∞ q pλ λ λ→∞ q pλ λ (1−p)δ + γ (1−p)δ + γ pλ λ q pλ λ q λ pλ λ (1−p)δ + γ + η (1−p)δ + γ − h (1−p)µ − (1−p)δ + γ = lim λ→∞ q pλ λ (1−p)δ + γ q 1 s (1−p)µ δγ = η − h = η − h , q p 1 µ(pγ + (1 − p)δ) (1−p)δ + γ
77 we obtain: s ! δγ P (Y ≤ n − s ) = Φ η − h , h = 0, .., ` (B.4) n h µ(pγ + (1 − p)δ) It follows from (B.2), (B.3), and (B.4) that
`−1 s ! X δγ lim ξ ≥ (Φ(β − h) − Φ(β − (h + 1)))Φ η − h λ→∞ µ(pγ + (1 − p)δ) h=0 s ! δγ + Φ(β − `)Φ η − ` (B.5) µ(pγ + (1 − p)δ) which is the lower Riemann-Stieltjes sum of the integral s ! Z ∞ δγ − Φ η + x dΦ(β − x) 0 µ(pγ + (1 − p)δ) s ! Z β δγ = Φ η + (β − t) dΦ(t) (B.6) −∞ µ(pγ + (1 − p)δ)
` corresponding to the partition {β − h}h=0 of the semi axis (−∞, β). Similarly, let us take the upper estimate for ξ as the following sum:
` sh i X X 1 λ − λ ξ ≤ ξ = e (1−p)µ · 2 i! (1 − p)µ h=0 i=sh+1 n−sh+1−1 j n−sh+1−j−1 k X 1 pλ − pλ X 1 λ − λ e (1−p)δ e γ j! (1 − p)δ k! γ j=0 k=0 (B.7) ` sh i X X 1 λ − λ = e (1−p)µ P (Y ≤ n − s − 1) i! (1 − p)µ n h+1 h=0 i=sh+1 ` X = P (sh+1 ≤ Xn ≤ sh)P (Yn ≤ n − sh+1 − 1) h=0 where Xn and Yn are the same random variable as before. Using the same calculation that were computed for the upper boundary we obtain `−1 s ! X δγ lim ξ ≤ (Φ(β − h) − Φ(β − (h + 1))) Φ η − (h + 1) λ→∞ µ(pγ + (1 − p)δ) h=0 (B.8) + Φ(β − `) which is the upper Riemann-Stieltjes sum for the integral (B.6). When → 0 the boundaries (B.5) and (B.8) lead to the following equality
s ! Z β δγ lim ξ = Φ η + (β − t) dΦ(t) λ→∞ −∞ µ(pγ + (1 − p)δ) This proves Lemma3.
78 B.4 Proof of Lemma4
Lemma 4. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED0 conditions. Define ζ as the expression
n−s−1 n−s−k−1 n−s−j−k−1 1 X X 1 X ζ = e−(RN +RD+RC ) R s R jR k ρi. (B.9) s! N j!k! D C k=0 j=0 i=0 Then s µ(pγ + (1 − p)δ) 1 lim ζ = √ (ηΦ(η) + φ(η)) . λ→∞ δγ 2π
Proof. First, we will rewrite Equation (B.9):
n−s−1 n−s−k−1 n−s−j−k−1 1 X X 1 X ζ = e−(RN +RD+RC ) R s R jR k ρi s! N j!k! D C k=0 j=0 i=0 n−s−1 n−s−k−1 1 X X 1 1 − ρn−s−j−k = e−(RN +RD+RC ) R s R jR k . s! N j!k! D C 1 − ρ k=0 j=0
When β = 0 by assumption QED0 (ii) ρ = 1 therefore,
n−s−j−k−1 X ρi = n − s − j − k. i=0
1−ρn−s−j−k When β → 0, ρ → 1 but still ρ 6= 1, the expression 1−ρ can be approximated by
1 − ρi ≈ i 1 − ρ thus
n−s−j−k−1 X lim ρi = n − s − j − k, ρ→1 i=0 which is the same phrase as when ρ = 1. Thus,
n−s−1 n−s−k−1 1 X X 1 ζ = e−(RN +RD+RC ) R s R jR k(n − s − j − k) s! N j!k! D C k=0 j=0 n−s−1 n−s−k−1 1 X X 1 = e−(RN +RD+RC ) R s (n − s) R kR j− s! N k!j! C D k=0 j=0
n−s−1 n−s−k−1 n−s−1 n−s−k−1 X k X 1 X 1 X j − R k R j − R k R j k! C j! D k! C j! D k=0 j=0 k=0 j=0 n−s−1 1 X 1 = e−(RN +RD+RC ) R s (n − s) (R + R )l− s! N l! C D l=0
79 n−s−2 n−s−k−2 n−s−2 n−s−k−2 X 1 X 1 X 1 X 1 − R R k R j − R R k R j C k! C j! D D k! C j! D k=0 j=0 k=0 j=0 n−s−1 1 X 1 = e−(RN +RD+RC ) R s (n − s) (R + R )l− s! N l! C D l=0 n−s−2 n−s−2 X 1 X 1 − R (R + R )l − R (R + R )l C l! C D D l! C D l=0 l=0 n−s−1 1 X 1 = e−(RN +RD+RC ) R s (n − s − R − R ) (R + R )l s! N C D l! C D l=0 (R + R )n−s + C D (n − s − 1)!
1 n−s−1 1 s−RN s X l −(RD+RC ) ≈ e √ ρ (n − s − RC − RD) (RC + RD) e 2πs l! l=0 ! (R + R )n−se−(RD+RC ) + C D (n − s − 1)!
1 n−s−1 1 X l −(RD+RC ) = √ (n − s − RC − RD) (RC + RD) e 2πs l! l=0 ! (R + R )n−se−(RD+RC ) + C D . (n − s − 1)! As seen in Equation (B.1)
n−s−1 X 1 p (n − s − R − R ) (R + R )le−(RD+RC ) ≈ η R + R Φ(η). (B.10) C D l! C D C D l=0 By using Stirling’s formula:
(R + R )n−se−(RD+RC ) (n − s)(R + R )n−se−(RD+RC ) C D = C D (n − s − 1)! (n − s)!
RC +RD n−s n−s−(RD+RC )+(n−s) ln (n − s)en−s−(RD+RC ) R + R (n − s)e n−s ≈ C D = p2π(n − s) n − s p2π(n − s) r n − s (n−s) 1− RD+RC +ln RC +RD = e n−s n−s . 2π
By assuming QED0 (i) when λ → ∞ R + R R + R (n − s) 1 − D C + ln C D n − s n − s ! R + R R + R 1 R + R 2 = (n − s) 1 − D C − 1 − C D − 1 − C D n − s n − s 2 n − s (B.11) n − s R + R 2 1 (n − s − R − R )2 = − 1 − C D = − C D 2 n − s 2 n − s √ √ 2 2 2 1 η RC + RD 1 η RC + RD η ≈ − √ ≈ − √ ≈ − . 2 η RC + RD + RC + RD 2 η RC + RD + RC + RD 2
80 Therefore, by assumption QED0 (i) √ r R +R R +R r 2 n − s (n−s) 1− D C +ln C D η RC + RD + RC + RD − η e n−s n−s ≈ e 2 2π 2π q p p = η RC + RD + RC + RDφ(η) ≈ RC + RDφ(η).
λ Combining the above approximations and the assumption that β = 0 and therefore s = RN = (1−p)µ yields
1 n−s−1 1 X l −(RD+RC ) ζ ≈ √ (n − s − RC − RD) (RC + RD) e 2πs l! l=0 ! (R + R )n−se−(RD+RC ) + C D (n − s − 1)! 1 p p ≈ √ η RC + RDΦ(η) + RC + RDφ(η) 2πs √ √ R + R R + R = √C D (ηΦ(η) + φ(η)) = √C D (ηΦ(η) + φ(η)) 2πs 2πRN rR + R 1 = C D √ (ηΦ(η) + φ(η)) RN 2π s (1 − p)µ pµ 1 = + √ (ηΦ(η) + φ(η)) . γ δ 2π
This proves Lemma4.
81 C Proof of approximation of the expected waiting time
In this appendix we will prove the approximation for the expected waiting time, stated in Section 5.2. The accurate measure was defined in Section 3.2, by Formula (3.5).
Theorem 4. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED6=0 condi- tions. Then √ φ(β)Φ(η) 1 φ( η2+β2) 1 η2 µ(pγ+(1−p)δ) q µ(pγ+(1−p)δ) 1 + e 2 1 Φ(η ) β − η − √ β β β 1 δγ δγ β lim sE[W ] = √ λ→∞ β q δγ φ(β)Φ(η) φ( η2+β2) 1 2 R 2 η1 µ −∞ Φ η + (β − t) µ(pγ+(1−p)δ) dΦ(t) + β − β e Φ(η1)
q µ(pγ+(1−p)δ) where η1 = η − β δγ .
Proof. It follows from (3.5) that the expectation of the waiting time is given by
n−1 l m Z ∞ 1 X X X E[W ] = p (s; t)dt = π (i, m − i, l − m)(i − s + 1) n µs n−1 0 l=s m=s i=s n−1 l m n−1 l m 1 X X X 1 X X X = π (i, m − i, l − m)(i − s) + π (i, m − i, l − m) µs n−1 µs n−1 l=s m=s i=s l=s m=s i=s (C.1) n−1 l m 1 X X X 1 = π (i, m − i, l − m)(i − s) + P (W > 0) µs n−1 µs l=s m=s i=s = C + D where D is given by
1 1 B D = P (W > 0) = µs µs A + B
(A and B where defined in Equation (5.1) and (5.2) respectively) and C is given by,
n−1 l m 1 X X X C = π (i, m − i, l − m)(i − s) µs n−1 l=s m=s i=s n−1 l m 1 X X X 1 1 = π (R )i (R )m−i (R )l−m (i − s) µs 0 s!si−s N (m − i)!(l − m)! D C l=s m=s i=s 1 G = . µs A + B
82 We will rewrite G in the following way:
n−1 l m X X X 1 1 G = (R )i (R )m−i (R )l−m (i − s) s!si−s N (m − i)!(l − m)! D C l=s m=s i=s n−s−1 n−s−k−1 n−j−k−1 X X X 1 1 = (R )i (R )j (R )k (i − s) s!si−s N j!k! D C k=0 j=0 i=s n−s−1 n−s−k−1 n−s−j−k−1 X X X i 1 = (R )i+s (R )j (R )k s!si N j!k! D C k=0 j=0 i=0 s n−s−1 n−s−k−1 n−s−j−k−1 (RN ) X X 1 X = (R )j (R )k i (ρ)i . s! j!k! D C k=0 j=0 i=0
Using the formula
0 M M ! 0 X X 1 − ρM+1 (−(M + 1)ρM )(1 − ρ) − (1 − ρM+1)(−1) lρl = ρ ρl = ρ = ρ = 1 − ρ (1 − ρ)2 l=0 l=0 (C.2) ρM+1 1 − ρM+1 ρM+1 1 − ρM = (M + 1) + ρ = ... = M + ρ ρ − 1 (1 − ρ)2 ρ − 1 (1 − ρ)2 one can rewrite G as a sum: G = G1 + G2; where
s n−s−1 n−s−k−1 n−s−j−k (RN ) X X 1 ρ G = (R )j (R )k (n − s − j − k − 1) 1 s! j!k! D C ρ − 1 k=0 j=0 and
s n−s−1 n−s−k−1 n−s−j−k−1 (RN ) X X 1 1 − ρ G = (R )j (R )k ρ . 2 s! j!k! D C (1 − ρ)2 k=0 j=0
Therefore,
s n−s−1 n−s−k−1 n−s−j−k (RN ) X X 1 ρ G = (R )j (R )k (n − s − j − k − 1) 1 s! j!k! D C ρ − 1 k=0 j=0 s n−s−1 n−s−k−1 n−s−j−k (RN ) X X 1 ρ = (R )j (R )k (n − s − 1) s! j!k! D C ρ − 1 k=0 j=0 s n−s−1 n−s−k−1 n−s−j−k (RN ) X X 1 ρ − (R )j (R )k (j + k) s! j!k! D C ρ − 1 k=0 j=0 s n−s n−s−1 n−s−k−1 j k (RN ) (n − s − 1) ρ X X 1 RD RC = s! ρ − 1 j!k! ρ ρ k=0 j=0 s n−s n−s−1 n−s−k−1 j k (RN ) ρ X X j + k RD RC − s! ρ − 1 j!k! ρ ρ k=0 j=0
83 s n−s n−s−1 l (RN ) (n − s − 1) ρ X 1 RD RC = + s! ρ − 1 l! ρ ρ l=0 s n−s n−s−1 l (RN ) ρ X l RD RC − + s! ρ − 1 l! ρ ρ l=0 s n−s n−s−1 l (RN ) (n − s − 1) ρ X 1 RD RC = + s! ρ − 1 l! ρ ρ l=0 s n−s n−s−2 l (RN ) ρ RD RC X l RD RC − + + s! ρ − 1 ρ ρ l! ρ ρ l=0 pλ pλ λ + + λ RD RC λ + + λ = −(n − s − 1)e (1−p)µ (1−p)δ γ ζ + + e (1−p)µ (1−p)δ γ ζ 2 ρ ρ 2 (R )s ρn−s 1 R R n−s + N D + C s! ρ − 1 (n − s − 1)! ρ ρ pλ pλ λ + + λ RD RC λ + + λ = −(n − s − 1)e (1−p)µ (1−p)δ γ ζ + + e (1−p)µ (1−p)δ γ ζ 2 ρ ρ 2 (R )s 1 n − s + N (R + R )n−s s! ρ − 1 (n − s)! D C where ζ2 was defined in (5.5); and
s n−s−1 n−s−k−1 n−s−j−k−1 (RN ) X X 1 1 − ρ G = (R )j (R )k ρ 2 s! j!k! D C (1 − ρ)2 k=0 j=0 s n−s−1 n−s−k−1 ρ (RN ) X X 1 = (R )j (R )k (1 − ρ)2 s! j!k! D C k=0 j=0 n−s s n−s−1 n−s−k−1 j k ρ (RN ) X X 1 RD RC − (1 − ρ)2 s! j!k! ρ ρ k=0 j=0 s n−s−1 ρ (RN ) X 1 = (R + R )l (1 − ρ)2 s! l! D C l=0 n−s s n−s−1 l ρ (RN ) X 1 RD RC − + (1 − ρ)2 s! l! ρ ρ l=0 ρ λ + pλ + λ 1 λ + pλ + λ = e (1−p)µ (1−p)δ γ ζ − e (1−p)µ (1−p)δ γ ζ (1 − ρ) 1 (1 − ρ) 2 1 λ + pλ + λ = e (1−p)µ (1−p)δ γ (ρζ − ζ ) (1 − ρ) 1 2
84 λ pλ λ − (1−p)µ + (1−p)δ + γ where ζ1 was defined in (5.4). Multiplying G by e we get
pλ − λ + + λ 1 RD RC Ge (1−p)µ (1−p)δ γ = (ρζ − ζ ) − (n − s − 1)ζ + + ζ (1 − ρ) 1 2 2 ρ ρ 2 s (RN ) 1 n − s − λ + pλ + λ + (R + R )n−s e (1−p)µ (1−p)δ γ s! ρ − 1 (n − s)! D C √ s β p RD RC ≈ 1 − √ ζ − ζ − η R + R + R + R − 1 ζ + + ζ β s 1 2 C D C D 2 ρ ρ 2 s (RN ) 1 n − s − λ + pλ + λ + (R + R )n−s e (1−p)µ (1−p)δ γ s! ρ − 1 (n − s)! D C (C.3) √ √ s s p 1 − ρ = ζ − 1 + ζ − − η R + R + (R + R ) − 1 1 β 2 β C D ρ C D s n−s n − s (RN ) (RD + RC ) + e−RN e−(RD+RC ) ρ − 1 s! (n − s)! r !! √ 1 RC + RD RC + RD 1 ≈ s ζ1 + ζ2 β − η − . β RN RN β
Due to the following approximation, we can neglect the second term: s (RN ) 1 n − s − λ + pλ + λ (R + R )n−s e (1−p)µ (1−p)δ γ s! ρ − 1 (n − s)! D C s (R ) 1 n − s λ pλ λ N n−s − (1−p)µ + (1−p)δ + γ ≈ (RD + RC ) e s! ρ − 1 p2π(n − s)e−(n−s)(n − s)n−s
s r n−s pλ (RN ) 1 n − s RD RC n−s− λ + + λ = + e (1−p)µ (1−p)δ γ s! ρ − 1 2π (n − s) (n − s)
s r R R (RN ) 1 n − s n−s− λ + pλ + λ +(n−s) ln D + C = e (1−p)µ (1−p)δ γ (n−s) (n−s) s! ρ − 1 2π s r R R (RN ) 1 n − s (n−s) 1− λ + pλ + λ +ln D + C = e (n−s)(1−p)µ (n−s)(1−p)δ (n−s)γ (n−s) (n−s) s! ρ − 1 2π √ s r 2 s 2 (RN ) 1 n − s −R − η n − s (RN ) −R 1 − η ≈ e N 2 = e N √ e 2 s! ρ − 1 2π ρ − 1 s! 2π √ RC + RD 1 p λ→∞ ≈ φ(RN + β RN )φ (η) → 0 ρ − 1 RN Where, R R R R R (n − s) 1 − N + C + D + ln D + C (n − s) (n − s) (n − s) (n − s) (n − s) R R R R R ≈ (n − s) 1 − N + C + D − 1 − D + C (n − s) (n − s) (n − s) (n − s) (n − s) ! 1 R R 2 − 1 − D + C 2 (n − s) (n − s) (n − s) R R 2 = −R − 1 − D + C N 2 (n − s) (n − s) η2 ≈ −R − N 2
85 (remark: for the last approximation see details in B.11) Combining the expressions for C and D we get
1 G 1 B 1 G + B 1 Ge−(RN +RD+RC ) + ζ E[W ] = C + D = + = = µs A + B µs A + B µs A + B µs ξ + ζ
−(R +R +R ) where ζ = ζ1 − ζ2. Thus, using the above approximation of Ge N D C (C.3), we get √ q 1 RC +RD RC +RD 1 √ 1 s ζ1 β + ζ2 R β − η R − β + ζ1 − ζ2 sE[W ] = √ N N µ s ξ + ζ1 − ζ2 q 1 RC +RD RC +RD 1 ζ1 β + ζ2 R β − η R − β (ζ − ζ ) = N N + √ 1 2 µ(ξ + ζ − ζ ) sµ(ξ + ζ − ζ ) 1 √2 1 2 2 2 1 2 q φ(β)Φ(η) 1 φ( η +β ) η RC +RD RC +RD 1 + e 2 1 Φ(η1) β − η − s→∞ β β β RN RN β → √ . q 2 2 1 2 R β RN φ(β)Φ(η) φ( η +β ) η µ Φ η + (β − t) dΦ(t) + − e 2 1 Φ(η1) −∞ RC +RD β β
The approximations to ζ1,ζ2 and ξ are stated in (5.6), (5.7) and (5.8), respectively.
The next theorem gives the approximation for the case where β = 0.
Theorem 5. Let the variables λ, s and n tend to ∞ simultaneously and satisfy the QED0 conditions. Then
µ(pγ+(1−p)δ) 2 √ 1 δγ (η + 1)Φ(η) + ηφ(η) lim sE[W ] = √ λ→∞ 2µ R 0 q δγ q µ(pγ+(1−p)δ) 2π −∞ Φ η − t µ(pγ+(1−p)δ) dΦ(t) + δγ (ηΦ(η) + φ(η))
q µ(pγ+(1−p)δ) where η1 = η − β δγ .
Proof. As before,
1 G + B E[W ] = µs A + B
We need to approximate G when β = 0.
s n−s−1 n−s−k−1 n−s−j−k−1 (RN ) X X 1 X G = (R )j (R )k i (ρ)i . s! j!k! D C k=0 j=0 i=0 Using the formula
M X (M + 1)(M) lρl = (C.4) 2 l=0 one can show that when β → 0, ρ → 1 but still ρ 6= 1, the sum used in formula (C.2) is approximately equal to the one stated in (C.4).
86 Thus, using the fact that s = RN , we get
s n−s−1 n−s−k−1 n−s−j−k−1 (RN ) X X 1 X G = (R )j (R )k i (ρ)i s! j!k! D C k=0 j=0 i=0 n−s−1 n−s−k−1 1 X X 1 (n − s − j − k)(n − s − j − k − 1) = (R )j (R )k s−ss! j!k! D C 2 k=0 j=0 using Stirling’s formula, and Lemma4 leading to
Ge−(RN +RC +RD) =
n−s−1 n−s−k−1 Rs X X 1 (n − s − j − k)(n − s − j − k − 1) = e−(RN +RC +RD) N (R )j (R )k s! j!k! D C 2 k=0 j=0 n−s−1 n−s−k−1 (n − s − 1) Rs X X 1 = e−(RN +RC +RD) N (R )j (R )k (n − s − j − k) 2 s! j!k! D C k=0 j=0 n−s−1 n−s−k−1 1 Rs X X (j + k) − e−(RN +RC +RD) N (R )j (R )k (n − s − j − k) 2 s! j!k! D C k=0 j=0 n−s−1 n−s−k−1 (n − s − 1) Rs X X 1 = e−(RN +RC +RD) N (R )j (R )k (n − s − j − k) 2 s! j!k! D C k=0 j=0 n−s−1 l 1 Rs X X l − e−(RN +RC +RD) N (R )j (R )l−j (n − s − l) 2 s! j!(l − j)! D C l=0 j=0 n−s−1 n−s−k−1 (n − s − 1) Rs X X 1 = e−(RN +RC +RD) N (R )j (R )k (n − s − j − k) 2 s! j!k! D C k=0 j=0 n−s−1 l 1 Rs X l X l! − e−(RN +RC +RD) N (n − s − l) (R )j (R )l−j 2 s! l! j!(l − j)! D C l=0 j=0 n−s−1 n−s−k−1 (n − s − 1) Rs X X 1 = e−(RN +RC +RD) N (R )j (R )k (n − s − j − k) 2 s! j!k! D C k=0 j=0 n−s−1 1 Rs X l − e−(RN +RC +RD) N (n − s − l)(R + R )l 2 s! l! D C l=0 n−s−1 n−s−k−1 (n − s − 1) Rs X X 1 = e−(RN +RC +RD) N (R )j (R )k (n − s − j − k) 2 s! j!k! D C k=0 j=0 n−s−2 1 Rs X 1 − e−(RN +RC +RD) N (R + R ) (n − s − l − 1) (R + R )l 2 s! D C l! D C l=0
87 n−s−1 n−s−k−1 (n − s − 1) Rs X X 1 = e−(RN +RC +RD) N (R )j (R )k (n − s − j − k) 2 s! j!k! D C k=0 j=0 n−s−1 1 Rs X 1 − e−(RN +RC +RD) N (R + R ) (n − s − l − 1) (R + R )l 2 s! D C l! D C l=0 n−s−1 n−s−k−1 (n − s − 1) Rs X X 1 = e−(RN +RC +RD) N (R )j (R )k (n − s − j − k) 2 s! j!k! D C k=0 j=0 n−s−1 1 Rs X 1 − e−(RN +RC +RD) N (R + R ) (n − s − l)(R + R )l 2 s! D C l! D C l=0 n−s−1 1 Rs X 1 + e−(RN +RC +RD) N (R + R ) (R + R )l 2 s! D C l! D C l=0 s n−s−1 n−s−k−1 (n − s − RD − RC − 1) R X X 1 = e−(RN +RC +RD) N (R )j (R )k (n − s − j − k) 2 s! j!k! D C k=0 j=0 n−s−1 1 Rs X 1 + e−(RN +RC +RD) N (R + R ) (R + R )l 2 s! D C l! D C l=0 s n−s−1 (n − s − RD − RC − 1) 1 R X 1 = ζ + e−RN N (R + R ) (R + R )l e−(RC +RD) 2 2 s! D C l! D C √ l=0 η RD + RC RD + RC 1 ≈ ζ + es−RN (ρ)s √ Φ(η) 2 2 2πs √ η R + R rR + R 1 R + R 1 ≈ D C C D √ (ηΦ(η) + φ(η)) + D√ C (ρ)s √ Φ(η) 2 RN 2π 2 RN 2π 1 R + R 1 R + R ≈ √ D√ C η2Φ(η) + ηφ(η) + √ D√ C (ρ)s Φ(η) 2π 2 R 2π 2 R N √ N 1 R + R s R + R ≈ √ D√ C (η2 + 1)Φ(η) + ηφ(η) = √ D C (η2 + 1)Φ(η) + ηφ(η) . 2π 2 RN 2 2π RN Therefore, using Lemmas3 and4 we get √ √s RD+RC 2 √ 1 G + B 1 R (η + 1)Φ(η) + ηφ(η) + ζ sE[W ] = √ = √ 2 2π N µ s A + B µ s ξ + ζ R +R 2 1 D C (η + 1)Φ(η) + ηφ(η) s→∞→ √ RN 0 q q 2 2π µ R Φ η − t RN dΦ(t) + RC +RD √1 (ηΦ(η) + φ(η)) −∞ RC +RD RN 2π RD+RC (η2 + 1)Φ(η) + ηφ(η) 1 RN = √ q q 2µ 2π R 0 Φ η − t RN dΦ(t) + RC +RD (ηΦ(η) + φ(η)) −∞ RC +RD RN µ(pγ+(1−p)δ) 2 1 δγ (η + 1)Φ(η) + ηφ(η) = √ . 2µ R 0 q δγ q µ(pγ+(1−p)δ) 2π −∞ Φ η − t µ(pγ+(1−p)δ) dΦ(t) + δγ (ηΦ(η) + φ(η))
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