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Queues in Hospitals: Semi-Open Queueing Networks in the QED Regime (QED = Quality- and Efficiency-Driven)

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Queues in Hospitals: Semi-Open Queueing Networks in the QED Regime (QED = Quality- and Efficiency-Driven) 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/1). 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 diagfag, a 2 R The matrix diag fa1; :::; aK g @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;
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