STOCHASTIC MODELS - Queueing theory
[email protected] 1 1 Queueing theory
• Queueing theory is the mathematical study of waiting lines, or queues. It finds origin thanks to A. K. Erlang, who modeled the Copenhagen telephone exchange.
• Finds applications in many areas: ICT, traffic, industrial and manufacturing engineering, shops, offices, hospitals, as well as in project management
• Enables the prediction of several QoS metrics of any client-server process, e.g., a) Mean time spent in the queue b) Mean queue length c) Mean server utilization factor d) Mean service time e) Throughput (or productivity)
[email protected] 2 Queues • A queue, or “queueing node” is a complex objected consisting of a) Many jobs or “customers” that arrive with a given rate b) One or more “servers” that, once free, can each be paired with an arriving job c) A “buffer” where costumers wait their turn, accordingly with a given policy, until a server is free to receive the job
Example 1: A queue with 3 servers a. is “idle”, thus an arrival is given to it to process b. is “busy” and will take time arrivals before complete the service waiting area c. has complete a service and is “free” to receive the next job. departure
[email protected] 3 Queues (cont’d) • The queuing theory helps the design and dimensioning of client-server systems • A single queue is described by a CT-BDP. It describes arrivals and departures in terms of the rates 휆푖 and 휇푖, and the number of costumers in the system.
휆0 휆1 휆3 0 1 2 3
휇1 휇2 휇3
• Remark: In practice it is of interest the study of “queueing networks”, namely a combination of queuing nodes, and the evaluation of the “customer routing”
• Example 2: The Internet is packet switching system. There many packets coming from different sources and with different destinations arrive to routers. • In the router they wait up to be routed through their destination.
[email protected] 4 Examples • Example 3: A router consist of 4 major components: a) Input ports b) Output ports c) switch fabric d) routing processor.
• Each I/O port maintains an I/O buffer
• The switch fabric moves packets from an input link to an output link
• The routing processor maintains the routing table and makes routing decisions
[email protected] 5 Markovian queueing nodes • A Markovian queue is characterized by having the independent arrivals and independent services pattern governed by different Poisson processes:
Arrival rate: 휆 →
Service rate: 휇 →
• Thus, the inter-arrival time and the service-times are exponentially distributed • Remark: Real queues are not memoryless, but often it is a good approximation
• Example 4: Assume a 푡0 = 0 the server is became free. Let Δ푡 = 푡 − 푡0, then
mean service time
[email protected] 6 Characteristics of queues and the Kendall Notation
• A single queueing node is usually described by using the Kendall Notation • It provides a compact description of a queue by means of a 6 fields string
A / B / m / K / N /
• Each field describes a different characteristics, respectively, ✓ A : arrival pattern ✓ B : service pattern serv. 1 arrival pattern serv. 2 ✓ m : number of servers «A» buffer ✓ K : buffer size «K» serv. m ✓ N : population size service pattern ✓ : queue’s discipline «B»
Remark: The last 3 fields are ommitted in the case of K = N = and = FIFO.
[email protected] 7 A : arrival pattern • The “arrival pattern” defines the way customers enter the resource (or queue)
• It describes the type of stochastic process 푋𝐴(푡), 푡 ∈ [0, ∞) associated with arrivals process, and how they occurs at random time-intervals
• It is often expressed in term of the inter-event time between 2 adjacent arrivals
• The “inter-arrival time” may be deterministic or stochastic
• In the Kendall notation 퐴 ∈ {퐷, 푀, 퐺, 퐸푘}, where:
✓ D : deterministic, e.g., Δ푡푘 = 1sec., ∀ 푘
✓ M : markovian, which implies Δ푡푘~퐸푥푝 휆 → Poisson distributed arrivals ✓ G : inter-arrival times with general distribution
✓ 퐸푘 : erlang with k as the shape parameter [email protected] 8 B : service pattern • The “service pattern” is often expressed in terms of its service rate 휇, namely, the mean number of consecutive services performed within a given time interval
• The “service time” may be deterministic or stochastic
• If the service process is Poisson distributed, then the service-time is exponential distributed, and the mean service time is
• In the Kendall notation it can be 퐴 ∈ {퐷, 푀, 퐺, 퐸푘}, where:
✓ D : deterministic, e.g., Δ푡푘 = 1sec., ∀ 푘
✓ M : markovian, which implies Δ푡푘~퐸푥푝 휆 ✓ G : inter-arrival times with general distribution
✓ 퐸푘 : erlang with 푘 as the shape parameter [email protected] 9 m : number of servers • The number of servers (or channels) defines the maximun number of costumers (or packets can be routed at the same time) by the resource
• Note that in any case, each server can only serve one user at a time.
• In the Kendall notation it can be ✓ 푚 = 1 : single server ✓ 푚 > 1 : multiple servers ✓ 푚 = ∞ : infinite servers
1° st. 2° st. • Example 6: A 2-stages load-balancing switch with 3 I/O-ports may enable the forwarding, at most, to 3 channels. Sequence of 3 matchings [email protected] 10 K : buffer size • The “buffer size” defines the maximum number of costumers can be accommodated in the waiting area. • Thus it accounts only the capacity of the waiting room.
• In the Kendall notation it can be: ✓ 퐾 = ℕ+ : finite capacity ✓ 퐾 = ∞ : infinite capacity
• Remark 1: The term “queue” or “queuing node” is refer to the considered resource, whereas the term “buffer” refers only to the waiting room • Remark 2: At most 퐾 + 푚 costumers can be in the resource at the same time. However However, sometimes it is denoted as 퐾 = 푚 + 푘 where 푘 is the buffer size, and 푚 is the number servers.
• Remark 3: If the buffer is full and a job arrives that job is rejected/discharged
[email protected] 11 N : population size
• The population size defines the maximun number of potential costumers
• A small population significantly affect the effective arrival rate, because, as more jobs queue up, there are fewer left available to arrive into the system.
• In the Kendall notation it can be: ✓ 푁 = ℕ+ : limited population ✓ 퐾 = ∞ : unlimited population
• Remark 1: It is a common assumption consider the population unlimited
• Remark 2: If 퐾 = ∞, this information is often omitted in the Kendall Notation
[email protected] 12 : queue’s discipline • The queue discipline or “service discipline” describe the under which priority order jobs in the waiting line are served • In the Kendall notation it can be:
Symbol Name Description Jobs are served in the order they arrived in (if FIFO “First In First Out” omitted is FIFO by default). Jobs are served in the reverse order to the order LIFO “Last in First Out” they arrived in. “Service In Random Jobs are served in a random order with no regard SIRO Order” to arrival order.
Many options: Preemptive Priority, Non PQ “Priority service” Preemptive, Class Based Weighted Fair Queuing, Weighted Fair Queuing PS “Processor Sharing”
[email protected] 13 Examples
• Example 7: Comparison between FIFO and Weighted Fair Queueing (WFQ). • The advantage of a FIFO is its simplicity and the fact that in many cases it properly describe the system needs. • It works properly when the population of jobs is sufficiently homogeneous. • When the jobs flow is not homogeneous, types of jobs may require different services. Thus a type of flow may affect the whole system performance. • In the Weighted Fair Queueing discipline jobs sorted into classes, and treated as a separate class and a separate queue is maintained for each class
FIFO Weighted Fair Queueing [email protected] 14 Quantity of interest of a queueing node • 푥 푡 ∈ 푋 = ℕ+ : the number of costumers in the queue at time 푡
• 휋푖 푡 ∈ 0,1 : probability of having exactly 푖 costumers in the queue at time 푡
• Π 푧, 푡 ∈ ℂ : Moment generating function associated with 푥(푡)
• 푥ҧ 푡 ∈ ℝ≥0 : Mean number of costumer in the queue at time 푡
+ • 푥푏 푡 ∈ ℕ : the number of buffered costumers in at time 푡
[email protected] 15 Examples • Example 8: Consider a single link shared by 푁 sources modelled as a DT-Queue with a single server and a buffer with infinite capacity. • The server can forward 퐶 packets per time slot + • Assume the number of packets 푎푖 푡 ∈ ℕ injected by sources IID and
• so the total arrival rate is less than the link capacity. • Due to the independence of sources the system can be modelled by a M/D/1 + • Moreover, let 푥푏 푡 ∈ ℕ be the number of buffered packet, one has that
[email protected] 16 Quantity of interest of a queueing node (cont’d)
• 휆 ∈ ℝ≥0 : arrival rate, i.e., the number of arrivals per unit of time.
• It follows that the mean inter-arrival time is 1/휆
• 휇 ∈ ℝ≥0: service rate, i.e., the mean number of services completed by the system within an unit of time.
• It follows that the mean service time is 1/휇
• 휃ҧ ∈ ℝ≥0: mean time spent in the queuing node
• 휃푏ҧ ∈ ℝ≥0: mean time spent in the waiting area
• … and others that we'll see later.
[email protected] 17 Little’s Law • The Little’s Law is a result of John Little which states that:
Little’s Law • Let 휆 be the effective long-term average arrival rate within the system • Let 휃ҧ be the average time spent by a customer in the system. • Let 푥ҧ be the the long-term average number of customers in the system • The Little’s Law says that
• Remark 1: This result a very general result. It is independent by either by the arrival, or the service distribution process, or by the priority discipline, or else
• Remark 2: It holds also for queuing networks, as long as they are not closed as we shall see later.
[email protected] 18 Intuitive explanation of the Little’s Law • Consider a queueing node operating in the steady regime • Consider a customer that is leaving the system after being served • If the costumer looks back, s/he will see an average a number of customers equal to the number of customers who arrived while s/he was in the system
• Remark: In the case of G/G/1 the Little’s Law further implies that:
• Example 9: A costumer spent 10 minutes in the system before his/her departure. • Let 휆= 4 costumers/5 minute 휆 buffer departure
[email protected] 19 Deterministic queues • Let the arrivals and service rate be constant, uniform, and deterministic as
• Operating hypothesis: If an arrival and a departure occurs simultaneously, the departure is occurred first. 푥(푡) 2 This prevents the buffer from being 1 occupied for measure zero time-instants 푡 (for queues with many servers is big deal)
• Remark: Due to the orderliness property that assumption is not necessary for a Markovian processes because two events can never occur at the same time. • To study the long term response of a D/D/⋅ queue we have to analyze 푥(푡) until a stationary, possibly periodic, response is observed (if this exists). • Let us now focus on the evolution of a D/D/1 queue.
[email protected] 20 D/D/1 • Consider a queue with deterministic constant arrivals and services rates ✓ If 휆 > 휇 the queue is unstable and lim 푥 푡 = ∞ 푡→∞ ✓ If 휆 < 휇 the queue is stable and lim 푥 푡 = 푥ҧ 푡→∞ ✓ If 휆 = 휇, thanks to the departures’ priority w.r.t. arrivals, the queue is stable
• Let us now focus on the queue with 휆 < 휇 where:
✓ 1st arrival at 푡 = 0 푥(푡) departure departure departure 1 ✓ 1st departure at t = 휇 1 1 푡 ✓ 2nd arrival at 푡 = 휆 1 1 1 1 2 2 1 3 0 + + 1 1 μ 휆 휆 μ 휆 휆 μ 휆 ✓ 2nd departure at 푡 = + arrival 휆 휇 arrival arrival arrival ✓ Etc… [email protected] 21 D/D/1 (cont’d) 푥(푡) departure departure departure
1
1 1 1 1 2 2 1 3 푡 0 + + μ 휆 휆 μ 휆 휆 μ 휆 arrival arrival arrival arrival • It is evident that 푥(푡) exhibits a periodic response with period 푇 = 1/휆
• The proportion of time the resource is busy, namely, the utilization factor, is
Utilization factor 0 < 푈 ≤ 1
• Since 휆 ≤ 휇 the buffer is not necessary, in particular, one has that
[email protected] 22 D/D/m • Consider a queue with deterministic constant arrivals and services rates ✓ If 휆 > 푚휇 the queue is unstable because the servers are always busy ✓ If 휆 ≤ 푚휇 the queue is stable thanks to the departure’s priority assumption
• In particular, let 휆 ≤ 푚휇, then it exists an integer 1 ≤ 푚ෝ ≤ 푚 such that
[email protected] 23 D/D/m (cont’d) • The long-term evolution of 푥(푡) is periodic with period 푇 = 1/휆 • Thus, at steady-state 푥(푡) may assume either 푚ෝ − 1 or 푚ෝ with probabilities
• From which, or or equivalently by the Little’s Law, one obtain:
[email protected] 24 D/D/m (cont’d) • Finally, the utilization factor, namely, the proportion of time a generic server is working (or equivalently the probability that a server is busy) is
Utilization factor of a D/D/m
D/D/∞
• In the case of a D/D/∞, by mean of the Little’s Law it can be proved that
• Because in a D/D/∞ there are ∞ servers and the number of busy servers is finite, and on average equal to 푥ҧ, so the utilization per server is zero.
There is always an Utilization factor of a D/D/∞ infinity of servers free
[email protected] 25 Traffic intensity • In telecommunication networks, traffic intensity measures the average occupancy of a shared resource during a specified period of time, namely,
It is equivalent to the utilization factor if the queue has a single server
• The traffic intensity can be seen also as the ration between the offered traffic and the carried traffic in a telecommunication network. • It is measured in Erlangs (traffic units).
• Example 10: In a digital network, the traffic intensity is:
✓ 푎 [packet/s] is the average arrival rate ✓ 퐿 [bit] is the average packet length ✓ 푅 [bit/s] is the transmission rate
[email protected] 26 Examples • Example 11: Consider a cellular network. Each channel can serve a phone call. • Assume that in this moment are 100 users in the system. • Each, on average, make one call of 3 minutes per hour. • Calculate the overall traffic intensity.
• Remark: If 휌 > 1 the queuing delay will grow without bound (if the traffic intensity stays the same). If 휌 < 1 the router can handle more average traffic
[email protected] 27 Traffic volume • In telecommunications, it is common refer also to the traffic volume, that is
• It is measured in erlang-hour (or erlang-minute, or call-hour or call-minute, etc…) and it is a measure of the traffic processed during a period of time.
• In telecommunication, service providers are vitally interested in traffic intensity and traffic volume, as it dictates the amount of equipment they must supply.
• Remark 1: If the traffic intensity 휌 > 1, then as we are going is the mean number of servers occupied in steady state by that traffic 휌, namely (푥ҧ푠 = 휌). • Remark 2: The quantity 휌 = 휆/휇 takes the meaning of traffic intensity only for M/M/1 and M/M/∞ queues. Not for multi-server resource, where there, 휌 expresses the traffic load to each server, namely 휌 = 휆/(푚휇).
[email protected] 28 Markovian queues • In most of practical applications, it is of interest considering queues which arrival process and the service processes are Poisson distributed • The following queues are of particular interest
✓ M/M/1 (classic resource) ✓ M/M/1 with discouraged arrivals ✓ M/M/1/K (with finite buffer) ✓ M/M/m (with m servers) ✓ M/M/∞ (with ∞ servers)
• In the following the conditions under which each queue reaches an unique stationary equilibrium independently from the initial state, namely, under which condition the queue is ergodic, will be discussed.
[email protected] 29 PASTA (Poisson Arrivals See Time Averages) Property • The PASTA property (or Arrival Theorem) refers to expected state of a queueing system seen by a new arrival from a Poisson Process. • PASTA Property: Because arrivals occur at random time instants. The probability that at time 푡 a costumer from outside find 푛 costumers in the system is equal to the steady state probability of finding 푛 costumers without that job.
휋푛 푡 = Pr 푥 푡 = 푛 = 휋푠,푛 = Pr 푥 ∞ = 푛 ∀ 푡 ≥ 0
• This implies that expected value of any parameter of the queue at the instant of a Poisson arrival is simply the long-run average value of that parameter.
• For example, at the instant of a Poisson arrival, it results that ✓ the mean number of customers in the queue is 푥ҧ = 휆 ⋅ 휃ҧ ✓ the probability that a server is busy is 푈 ✓ the probability that a server is in idle is 1 − 푈
✓ the mean number of buffered costumers is 푥ҧ푏 = 푥ҧ − 푈 [email protected] 30 M/M/1 • A M/M/1 describe queuing node with a single server, Poisson distributed arrivals with rate 휆 and exponential service times with rate 휇.
• A M/M/1 is described by a uniform time-homogeneous CT-BDP 휆 휆 휆 0 1 2 3 휇 휇 휇 • If 휆 and 휇 are constant and state-independent this MC is aperiodic and irreducible • However, to be be ergodic, it is required that
[email protected] 31 M/M/1 : Quantities of interest • By exploiting the results shown for CT-BDPs, one has that:
• Probability to find 푖 costumers in the system is
• Idle rate, namely, the probability the system is empty
• Utilization factor of the resource (and in this case of the server), namely, the average occupancy of a resource during a specified period of time is
[email protected] 32 M/M/1 : Quantities of interest (cont’d) • Burke's theorem: If arrivals are Poisson distributed with rate 휆, then at steady state the departure process is Poisson as well with the same rate 휆.
• From the Burke's theorem it results that the throughput, namely, the productivity of the resource at steady state is
• Since the service time is exponentially distributed, the mean service time at steady state remains constant and equal to the inverse of the nominal rate
• Mean number of costumer in the resource at steady state
Hint: Use Π 푧, 푡 or
[email protected] 33 M/M/1 : Quantities of interest (cont’d) • From the Little’s Law the mean time spent in the system by a customer is
• Mean number of buffered costumers is: • From a single queue system one has
• Thus
• Mean number of busy servers, which coincides with the utilization factor, is
[email protected] 34 M/M/1 : Quantities of interest (cont’d)
• The utilization of a single server at steady state, since 푚 = 1, becomes
• Mean time spent in the buffer by a costumer
• Remark: Note that as the traffic intensity 휌→ 1, then 푥ҧ, and 휃ҧ, and 휃푏ҧ → .
[email protected] 35 Multiplexing • In telecommunications, the multiplexing enable multiple traffic streams from possibly different sources to share a common transmission resource.
• Operating assumption: Each source generates packets in a Poisson fashion, and the packet lengths is exponentially distributed.
• Then by means of a M/M/1 the multiplexing performance can be evaluated!
• Let us first note in a M/M/1, the queue-size probabilities depends only by 휌, whereas the delay statistics 휃ҧ, and 휃푏ҧ depends on the inverse of 휇 − 휆, e.g.,
• It follows that if both the arrival and service rate increases to 푁 the state probabilities are the same, whereas
[email protected] 36 Multiplexing: Time division multiple access (TDMA) • Consider the TDMA multiple access techniques, where each user is assigned to one or more channels (in a form of time-slots) to access the shared link. • Sources generated their own traffic with rate λ [bit/sec] • In the TDMA sources transmits only within their allocated time-slots with a rate µ [bit/sec], meanwhile the other traffic is buffered in their own buffer.
• It results that, for each source behaves as a M/M/1 such that
[email protected] 37 Multiplexing: Full multiplexing (FMUX) • Consider the FMUX (or statistical multiplexing) access techniques. • All users send their own generated traffic, which rate is 휆, to a switch, so the switch’s arrival rate is 푁휆 • The switch forwards the data to the destination with a rate that is 푁휇. • The difference is that now the traffic of all the sources is stored in one buffer
• In the FMUX sources send their data to the switch which behaves as a M/M/1
[email protected] 38 Multiplexing comparison
Average Packet Service Time: TDMA vs FMUX
20
20
[email protected] 39 Multiplexing dimensioning • Let the QoS measure of interest be the mean delay 휃ҧ • Assume the packets lengths being exponentially distributed and the packet generation process Poisson distributed • Assume that for the nominal traffic intensity 휌 our specification is met
• If then, the arrival rate increases to 푁휆, we aim to find 휇∗ ∶ 휃ҧ = 휃ҧ∗
푁휆 • Now 휌∗ = > 휌, thus increasing 푈, while reducing Pr 푥 = 0 = 1 − 휌∗. 휇∗
• Example 13: A service provider wishes to migrate to a dynamic statistical access mechanism. However the QoS of TDMA was satisfactory. • Then it does not need to allocate the total capacity, but 휇∗ = 휇 + 푁 − 1 휆
[email protected] 40 M/M/1 dimensioning based on the Delay Distribution • The previous dimensioning was focused only on the mean delay 휃ҧ only, independently to its probability distribution. • Sometimes it is more practical refers to a percentile of the delay distribution 휃ҧ. • Example 14: Requires that no more than 훼 = 1% of the packets will experience over Δ푡 = 100 millisecond of delay
• In the context of the M/M/1 model, we define 2 dimensioning problems • Problem 1: Notice first that because arrival and services are Poissonian the time spent in the resource 휃 is exponentially distributed with rate 1/휃ҧ
• For a given 훼 ∈ [0,1], Δ푡 > 0, we aim to find the minimal µ∗ such that
Solution:
[email protected] 41 M/M/1 dimensioning based on the Delay Distribution (cont’d)
• Problem 2: For a given 휇, 훼 ∈ [0,1], Δ푡 > 0, we aim to find the maximal 휆∗ such that
Solution:
• Remark: The solution 휆∗ to be feasible must satisfy that
[email protected] 42 M/M/1 with discouraged arrivals 휂 • The flow is subject to a 푖푛 휇 splitting process 푎푏 • 푎푏 is the abandonment rate • In this type resource the incoming costumers follows a Poisson fashion
• with a rate 푖푛 that depends on the actual number 푖 of customer in the system
• The service time are, as usual, IID and exponentially distributed with rate 휇 • It can be modelled by a time-homogenous, but not non-uniform, CT-BDP where the births decrease according to the hyperbolic law, and deaths are uniform
휆1 = 휆2 = /2 휆3 = /3
0 휇 1 휇 2 휇 3
[email protected] 43 M/M/1 with discouraged arrivals (cont’d) • To have the CT-BDP ergodic it is sufficient that
• The steady state probabilities can be derived by solving
where
• Then,
[email protected] 44 M/M/1 with discouraged arrivals (cont’d)
Idle probability
• The steady state probabilities can be derived by solving
[email protected] 45 M/M/1 with discouraged arrivals : Quantities of interest
• Probability of having exactly 푖 costumers in the system at steady state is
• Idle rate, namely, the probability that no costumer in the system
• Utilization factor, namely, the average occupancy of the resource during a specified period of time
[email protected] 46 M/M/1 with discouraged arrivals : Quantities of interest
• Throughput (or departure flow) at steady state, namely, the long term resource productivity rate
• Rate of arrivals within the resource at steady state • From the Burke's theorem 휂 at steady state 휂 = 휆푖푛 푖푛 휇
푎푏
• Rate of abandonment at steady state
[email protected] 47 M/M/1 with discouraged arrivals : Quantities of interest
• By mean of the moment generating function, it follows
• Mean number of costumer in the resource at steady state
• Remark: Notice that as the traffic intensity increase, the mean number of costumer in the system increase as well. However it cannot be greater than 1.
[email protected] 48 M/M/1 : Quantities of interest (cont’d) • From the Little’s Law the mean time spent in the system by a customer is
• From the Little’s Law the mean number of busy servers at steady state, which coincides with the utilization factor
• Mean number of buffered costumers at steady state
[email protected] 49 M/M/1 : Quantities of interest (cont’d)
• Mean time spent in the buffer by a costumer at steady state
• The single server utilization at steady state, since 푚 = 1, becomes
• Remark: The single server utilization is equal to the resource utilization factor because 푚 = 1.
[email protected] 50 M/M/1/K
휂 푖푛 휇
퐾 − 1 푎푏
• The M/M/1/K is a single server resource with a buffer with limited size.
• Costumers arrive in a Poisson fashion with a rate 휆, and the service times are, as usual, exponentially distributed with mean rate 휇
• Since the buffer is limited, a portion of arrivals may be lost due to overflow.
• Remark: In the Kendall notation, K denote the buffer size only. However, here for description purposes, K denotes the overall number of costumer in the system, thus the buffer has size equal to 퐾 − 1.
[email protected] 51 M/M/1/K (cont’d)
0 1 2 K-1 K • The M/M/1/K can be modelled by a finite-state CT-BDP a) the birth rate is state dependent and decreases as
b) The death rate 휇 is constant
• Since, the CT-BDP has a finite number of states, to be ergodic,
No more required that
[email protected] 52 M/M/1/K (cont’d) • Since the system is ergodic, we can evaluates its limiting distribution by solving the following linear system
[email protected] 53 M/M/1/K : Quantities of interest
• Probability of having exactly 푖 costumers in the system at steady state is
• The Idle rate, namely, the probability that no costumer in the system
• The blocking probability, namely, the proportion of incoming costumers that are blocked due to buffer overflow is (cfr. with 휆푎푏 = 휆 ⋅ 휋푠,푘)
[email protected] 54 M/M/1/K : Quantities of interest
• Utilization factor, namely, the average occupancy of the resource during a specified period of time
• Throughput (or departure flow) at steady state, namely, the long term productivity of a server is
• Mean arrivals rate within the resource at steady state
• From the Burke's theorem at steady state 휂 = 휆푖푛
[email protected] 55 M/M/1/K : Quantities of interest • Abandonment rate at steady state
• That is clearly proportional to the blocking probability
• By means of the moment generating function one has
Examples:
• Mean number of costumer in the resource at steady state
[email protected] 56 M/M/1/K : Quantities of interest • From the Little’s Law the steady state time spent in the system by a customer is
• The mean service time at steady state remains constant and equal to
• The mean time spent in the buffer by a costumer at steady state
[email protected] 57 M/M/1/K : Quantities of interest • Mean number of busy servers at steady state, which coincides with the utilization factor
• Mean number of buffered costumers at steady state
• The utilization of a single server at steady state, since 푚 = 1, becomes
[email protected] 58 M/M/m • The M/M/m is the generalization with 푚 independent servers of a M/M/1
• Since the queue-size process increases/decreases by only one, so it can be modeled by a CT-BDP where: • 1. The birth rate is constant
• 2. The death rate depends on the number of costumer in the resource
• where denotes the service rate.
0 1 m-1 m m+1 2 (m-1) m m m
[email protected] 59 M/M/m (cont’d) • Since the CT-BDP has an infinite capacity, the process is ergodic, if when all the server are busy, the service rate is sufficient to serve all the incoming costumer
• If the ergodic condition is satisfied, its limiting distribution is the solution of
[email protected] 60 M/M/m (cont’d)
• Remark 1: If it was considered a M/M/m/m, instead of a M/M/m, 휋푠,푖 = 0, ∀ 푖 > 푚 • Remark 2: For an M/M/m/m, the blocking probability 휋푠,푚 is called Erlang-B loss formula [email protected] 61 M/M/m : Quantities of interest
• The Idle rate, namely, the probability that no costumer in the system
• Probability of having exactly 푖 costumers in the system at steady state is
• Probability that all servers are busy, or equivalently, proportion of time an incoming costumers will have to wait before being served
Erlang-C Formula
[email protected] 62 M/M/m : Quantities of interest (cont’d) • The mean service time at steady state remains constant and equal to
• Mean number of busy servers at steady state Little’s Law
• The utilization of a single server at steady state becomes
• Throughput at steady state, namely, the long term productivity of a server is
(from the Burke's theorem)
[email protected] 63 M/M/m : Quantities of interest (cont’d)
• Mean number of costumer in the resource at steady state
(given without proof)
• Mean number of costumer in the buffer at steady state
• Mean time spent in the system by a customer at steady state
(from the Little’s Law)
• Mean time spent in the buffer by a costumer at steady state
[email protected] 64 M/M/∞ • Because the number of servers is unlimited all customers go immediately into the service area upon their arrival • The M/M/∞ can be modelled by CT-BDP as follows
0 1 2 3 2 3 4
1) Costumer arrive in a Poisson fashion with constant rate
2) The death rate increases with the number of busy servers
Clearly ergodic ∀ 휇 > 0, ∀ 휆 > 0 where 휇 denote the service rate of each server
[email protected] 65 M/M/∞ (cont’d) • Because ergodic its limiting distribution is the solution of
[email protected] 66 M/M/∞ (cont’d) • Remark 1: Althought the steady state probability of a M/M/∞ resource coincides with that of a M/M/1 with discouraged arrivals its functioning is completely different (cfr. with slide 45).
휆1 = 휆2 = /2 휆3 = /3
0 휇 1 휇 2 휇 3
• This fact would emphasize the following aspect:
The probability distribution of a given resource, seen individually, is not representative of how the resource works.
To fully characterize the queue completely we have instead consider the all the fields of the associated Kendall Notation
[email protected] 67 M/M/∞ : Quantities of interest
• Probability of having exactly 푖 costumers in the system at steady state is
• Idle rate, namely, the probability that no costumer in the system
• Utilization factor, namely, the average occupancy of the resource during a specified period of time
[email protected] 68 M/M/∞ : Quantities of interest (cont’d)
• Mean number of costumer in the resource at steady state (see slides 48)
• Remark: Notice that as the traffic intensity increase, the mean number of costumer in the system increase as well.
• Throughput at steady state, namely, the long term resource productivity (From the Burke's theorem)
• From the Little’s Law the mean time spent in the system by a customer is
• The mean service time at steady state remains constant and equal to
[email protected] 69 M/M/∞ : Quantities of interest (cont’d) • Mean time spent by a customer in the buffer
• Mean number of costumer in the buffer at steady state
• Mean number of busy servers at steady state
• The utilization of a single server at steady state, since 푚 = 1, becomes