Queueing Theory I. Summary. Little s Law Queueing System Notation Stationary Analysis of Elementary Queueing Systems. M/M/1 M/M/m M/M/1/K

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1 Queueing Theory I Suary Little s Law Queueing Syste Notation Stationary Analysis of Eleentary Queueing Systes M/M/ M/M/ M/M//

2 Little s Law a(t): the process that counts the nuber of arrivals up to t. d(t): the process that counts the nuber of departures up to t. N(t) a(t)- d(t) a(t) d(t) N(t) Area γ(t) Tie t Average arrival rate (up to t) t a(t)/t Average tie each custoer spends in the syste T t γ(t)/a(t) Average nuber in the syste N t γ(t)/t Little s Law d(t) a(t) N(t) Area γ(t) N T t t t Taking the liit as t goes to infinity Expected nuber of custoers in the syste E[ N ] E[ T ] Tie t Expected tie in the syste Arrival rate IN the syste

3 Generality of Little s Law E[ N ] E[ T ] Little s Law is a pretty general result It does not depend on the arrival process distribution It does not depend on the service process distribution It does not depend on the nuber of servers and buffers in the syste. Aggregate Arrival rate Queueing Network Specification of Queueing Systes Custoer arrival and service stochastic odels Structural Paraeters Nuber of servers Storage capacity Operating policies Custoer class differentiation (are all custoers treated the sae or do soe have priority over others?) Scheduling/Queueing policies (which custoer is served next) Adission policies (which/when custoers are aditted)

4 Queueing Syste Notation Arrival Process M: Markovian D: Deterinistic Er: Erlang G: General A/B///N Service Process M: Markovian D: Deterinistic Er: Erlang G: General Nuber of servers,, Storage Capacity,, (if then it is oitted) Nuber of custoers N,, (for closed networks otherwise it is oitted) Perforance Measures of Interest We are interested in steady state behavior Even though it is possible to pursue transient results, it is a significantly ore difficult task. E[S] average syste tie (average tie spent in the syste) E[W] average waiting tie (average tie spent waiting in queue(s)) E[X] average queue length E[U] average utilization (fraction of tie that the resources are being used) E[R] average throughput (rate that custoers leave the syste) E[L] average custoer loss (rate that custoers are lost or probability that a custoer is lost)

5 Recall the Birth-Death Chain Exaple At steady state, we obtain + In general... ( + ) Making the su equal to Solution exists if S + <... + M/M/ Exaple Meaning: Poisson Arrivals, exponentially distributed service ties, one server and infinite capacity buffer. - Using the birth-death result and, we obtain,,,,... Therefore + for / < ( ),,,...

6 M/M/ Perforance Metrics Server Utilization [ ] ( ) E U Throughput E R [ ] ( ) Expected Queue Length d { } E[ X ] ( ) ( ) d d d ( ) ( ) d d ( ) ( ) M/M/ Perforance Metrics Average Syste Tie E[ X] E[ S] E[ S] E[ X] E [ S] ( ) ( ) Average waiting tie in queue E[ S] E[ W] + E[ Z ] E[ W] E[ S] E[ Z ] E [ W ] ( ) ( )

7 M/M/ Perforance Metrics Exaples.5 4 Delay (tie units) / Nuber of custoers Ε[Χ] Ε[S] Ε[W] rho PASTA Property PASTA: Poisson Arrivals See Tie Averages Let (t) Pr{ Syste state X(t) } Let a (t) Pr{ Arriving custoer at t finds X(t) } In general (t) a (t)! Suppose a D/D/ syste with interarrival ties equal to and service ties equal to.5 a a a a Thus (t).5 and (t).5 while a (t) and a (t)!

8 Theore For a queueing syste, when the arrival process is Poisson and independent of the service process then, the probability that an arriving custoer finds custoers in the syste is equal to the probability that the syste is at state. In other words, a () t () t Pr X () t,,,... Proof: () li Pr () (, + Δ ) Δ t { } { t } a t X a tt t li Δ t { X () t a( tt+ Δt) } Pr { a ( tt, +Δt) } { X () t } a( tt+δt) Pr a( tt, +Δt) Pr,, { } { } Arrival occurs in interval (t,δt) Pr Pr, li Pr{ X () t } () t Δ t M/M/ Queueing Syste Meaning: Poisson Arrivals, exponentially distributed service ties, identical servers and infinite capacity buffer. 3 if and if +

9 M/M/ Queueing Syste Using the general birth-death result, if <!!, if Letting /() we get ( ) if <! if! To find ( ) + +!! + + ( ) ( )!! ( ) M/M/ Perforance Metrics Server Utilization ( ) E [ U ] + Pr{ X } +!! ( ) ( ) ( ) + + ( )!! ( ) ( ) ( ) ( ) ( ) ( )! ( )! ( )!! ( ) + + ( ) ( )!! ( )

10 M/M/ Perforance Metrics Throughput E[ R] + Expected Queue Length ( ) E[ X ]... +!! ( ) E[ X ] +! ( ) Using Little s Law ( ) E[ S] E[ X ] +! ( ) Average Waiting tie in queue E[ W] E[ S] M/M/ Perforance Metrics Queueing Probability PQ Pr{ X }!! ( ) ( ) Erlang C Forula

11 Exaple Suppose that custoers arrive according to a Poisson process with rate. You are given the following two options, Install a single server with processing capacity.5 Install two identical servers with processing capacity.75 and 3.75 Split the incoing traffic to two queues each with probability.5 and have.75 and 3.75 serve each queue. Α Β 3 C 3 Exaple Throughput It is easy to see that all three systes have the sae throughput E[R A ] E[R B ] E[R C ] Server Utilization [ ] E U A [ ] E U B [ ] EU C Therefore, each server is /3 utilized Therefore, all servers are siilarly loaded.

12 Exaple Probability of being idle A 3 Β C ( ) ( )!! ( ) For each server Exaple Queue length and delay [ ] E X A E [ X ] B.5 ( ) +! 5 ( ) E E A [ S ] E[ X ] A B 5 [ S ] E[ X ] B [ ] E X C /.5 /.75.5 [ ] E[ X ] E X 4 C C For each queue! C 4 [ ] E[ X ] E X C

13 M/M/ Queueing Syste Special case of the M/M/ syste with going to 3 + (+) and for all Let / then, the state probabilities are given by e + e!!! Syste Utilization and Throughput E[ U ] e E[ R] M/M/ Perforance Metrics Expected Nuber in the Syste E[ X ] e e!! Nuber of busy servers ( ) Using Little s Law [ ] [ ] E S E X No queueing!

14 M/M// Finite Buffer Capacity Meaning: Poisson Arrivals, exponentially distributed service ties, one server and finite capacity buffer. - Using the birth-death result and, we obtain,,,,... Therefore + + for / ( ),,,... + M/M// Perforance Metrics Server Utilization E [ U ] + + Throughput E [ R] ( ) < + Blocking Probability ( ) ( ) ( ) P B + Probability that an arriving custoer finds the queue full (at state )

15 M/M// Perforance Metrics Expected Queue Length ( ) ( ) { } d E[ X ] + + d ( ) d ( ) + d + + d d ( ) + ( ) ( ) ( )( + ) + ( ) + Syste tie E[ X ] ( ) E[ S] Net arrival rate (no losses) M/M// Queueing Syste Meaning: Poisson Arrivals, exponentially distributed service ties, servers and no storage capacity. 3 (-) - Using the birth-death result and, we obtain,,,,...! Therefore!! for /,,,...!

16 M/M// Perforance Metrics Blocking Probability /! PB! Erlang B Forula Probability that an arriving custoer finds all servers busy (at state ) Throughput /! <! [ ] ( ) E R M/M///N Closed Queueing Syste Meaning: Poisson Arrivals, exponentially distributed service ties, one server and the nuber of custoers are fixed to N. N N (N-) Using the birth-death result, we obtain N!,,,... N N! (N-) ( ) N N! ( N )! N- N

17 M/M///N Closed Queueing Syste N Response Tie (N-) Tie fro the oent the custoer entered the queue until it received service. For the queue, using Little s law we get, E[ X ] ( ) E[ S] (N-) N- In the thinking part, E[ N X] ( ) Therefore N ( ) N E [ S] ( ) ( ) N