Queueing Theory II

1. Queueing Theory II

2. Summary • M/M/1 Output process • Networks of Queue • Method of Stages • Erlang Distribution • Hyperexponential Distribution • General Distributions • Embedded Markov Chains

3. M/M/1 Output Process Ak Ik Zk Zk Yk Yk Interdeparture Time Dk-1 Dk-1 Dk Dk • Burke’s Theorem: • The Departure process of a stable M/M/1 queueing system with arrival rate λ is also a Poisson process with rate λ.

4. Burke’s Theorem PASTA PASTA

5. Two Queues in Series λ μ1 μ2 λ 02 12 μ2 μ2 λ λ 01 11 21 μ2 μ2 μ2 μ1 μ1 μ1 μ1 μ1 λ λ λ 00 10 20 30 • Let the state of this system be (X1,X2) where Xi is the number of customers in queue i.

6. Two Queues in Series λ 02 12 μ2 μ2 λ λ 01 11 21 μ2 μ2 μ2 μ1 μ1 μ1 μ1 μ1 λ λ λ 00 10 20 30 • Balance Equations

7. Product Form Solution • Let ρ1=λ/μ1, ρ2=λ/μ2 • Recall the M/M/1 Results • Therefore the two queues can be “decoupled” and studied in isolation.

8. Jackson Networks • The product form decomposition holds for all open queueing networks with Poisson input processes that do not include feedback • Even though customer feedback causes the total input process (external Poisson and feedback) become non-Poisson, the product form solution still holds! • These types on networks are referred to as Jackson Networks • The total input rate to each note is given by External Arrival Rate Aggregate Rate Routing Prob

9. Closed Networks • The product form decomposition holds for also for closed networks Aggregate Rate Routing Prob

10. Non-Poisson Processes Assume that the service time distribution can be approximated by the sum of m iid exponential random variables with rate mμ. That is • In Homework 4, you showed the density of Z is given by • Z is an Erlang random variable with parameters (m,μ).

11. Erlang Distribution You can show that the distribution function of Z • The expected value of Z is given by • The variance of Z is given by • Note that the variance of an Erlang is always less than or equal to the variance of the exponential random variable

12. M/Erm/1 Queueing System mμ mμ mμ Erlang Server Stage 2 Stage 1 Stage m • Meaning: Poisson Arrivals, Erlang distributed service times (order m), single server and infinite capacity buffer. • Only a single customer is allowed in the server at any given time. • The customer has to go through all m stages before it is released.

13. M/Erm/1 Queueing System λ λ λ λ λ λ λ 0 1 2 3 4 n-1 n 2μ 2μ 2μ 2μ 2μ 2μ 2μ 2μ • The state of the system needs to take into account the stage that the customer is in, so one could use (x,s) where x=0,1,2… is the number of customers and s=1,…,m is the stage that the customer in service is currently in. • Alternatively, on can use a single variable y to be the total number of stages that need to be completed before the system empties. • Let m=2

14. Erm/Μ/1 Queueing System 2λ 2λ 2λ 2λ 2λ 2λ 2λ 2λ μ μ μ μ μ μ μ 0 1 2 3 4 n-1 n • Similar to the M/Erm/1 System we can let the state be the number of arrival stages in the system, so for example, for m=2

15. Hyperexponential Distribution μ1 p1 p2 μ2 pm Hyperexponential Server μm • Recall that the variance of the Erlang distribution is always less than or equal than the variance of the exponential distribution. • What if we need service times with higher variance?

16. Hyperexponential Distribution • Expected values if the Hyperexponential distribution • The variance of Z is given by

17. M/Hm/1 Queueing System 11 21 31 n1 λp pμ1 pμ1 pμ1 pμ1 μ1 (1-p)μ1 (1-p)μ1 (1-p)μ1 0 pμ2 pμ2 pμ2 μ2 (1-p)μ2 (1-p)μ2 (1-p)μ2 12 22 32 n2 λ(1-p) λ λ λ λ • In this case, the state of the system should include both, the number of customers in the system and the stage that the customer in service is in. For example, for m=2 λ λ λ λ (1-p)μ2

18. M/G/1 Queueing System • When the arrival and service processes do not possess the memoryless property, we are back to the GSMP framework. • In general, we will need to keep track of the age or residual lifetime of each event. • Embedded Markov Chains • Some times it may be possible to identify specific points such that the Markov property holds • For example, for the M/G/1 system, suppose that we choose to observe the state of the system exactly after each customer departure. • For this problem, it doesn’t matter how long ago the previous arrival occurred since arrivals are generated from Poisson processes. Furthermore, at the point of a departure, we know that the age of the event is always 0.

19. Embedded Markov Chains a1 ak-n+1 a3 a2 n-2 n-1 n n+1 n+2 k a0 • So, right after each departure we observe the following chain (only the transitions from state n are drawn) • Let aj be the probability that j arrivals will occur during the interval Y defined by two consecutive departures • Let N be a random variable that indicates the number of arrivals during Y, then aj = Pr{N = j}.

20. Embedded Markov Chains • Suppose we are given the density of Y, fY(y) then • Since we have a Poisson process • Therefore

21. State Iteration Xk-1 Xk Xk Xk-1 Ak Ik Zk Zk Yk Yk Dk-1 Dk-1 Dk Dk • Let Xk be the state of the system just after the departure of the kth customer Arrivals DuringYk

22. M/G/1 Queueing System • Pollaczek-Khinchin (PK) Formula where • X is the number of customers in the system • 1/μis the average service time • σ2is the variance of the service time distribution • λ is the Poisson arrival rate • ρ=λ/μis the traffic intensity

23. M/G/1 Example • Consider the queueing systems M/M/1 and M/D/1 • Compare their average number in the system and average system delay when the arrival is Poisson with rate λ and the mean service time is 1/μ. • For the M/M/1 system, σ2=1/μ2, therefore • For the M/D/1 system, σ2=0, therefore