On the variance curve of outputs for some queues and networks Yoni Nazarathy Gideon Weiss Yoav Kerner CWI Amsterdam March 2009

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3 Queueing System Output Counts Example 1: Stationary stable M/M/1, D(t) is PoissonProcess( ): Example 2: Stationary M/M/1/1 with. D(t) is RenewalProcess(Erlang(2, )): Asymptotic Variance Rate Y-intercept

4 Finite Capacity Birth-Death Queues

5 Scope: Finite, irreducible, stationary, birth-death CTMC that represents a queue.

6 Theorem Part (i) Part (ii) Scope: Finite, irreducible, stationary, birth-death CTMC that represents a queue. and If Then Calculation of (Asymptotic Variance Rate of Output Process)

7 Proof Outline 1) Look at M(t)=D(t)+E(t) instead of D(t). 2) Proposition: The “Fully Counting” MAP of M(t) has associated MMPP with same variance. 3) Results of Ward Whitt: An explicit expression for the asymptotic variance rate of birth-death MMPP.

8 M/M/1/K “BRAVO Effect”

9 01 K K – 1 Some intuition for M/M/1/K-BRAVO …

10 M/M/40/40 M/M/10/10 M/M/1/40 K=20 K=30 c=30 c=20 Other Birth-Death Queues (M/M/c/K)

11 MAP used for PH/PH/1/40 with Erlang and Hyper-Exp distributions General Processing Times

12 The “ 2/3 property ” GI/G/1/K SCV of arrival = SCV of service

13 For Large K Covariance Between Counts

14 General Lossless Queues

15 Stable Lossless Queues Preserve Asymptotic Variance Proof for stable case:

16 M/G/1 Queue

17 M/G/1 Linear Asymptote Theorem:

18 Derivation Method: Embedding in Renewal Reward Busy Cycle Duration Number Customers Served

19 Linear Asymptote of Renewal Reward is Known Brown, Solomon 1975:

20 Using in Regenerative Simulation

21 Naive Estimation of Asymptotic Variance: There is bias due to intercept: Regenerative Estimation of Asymptotic Variance: Estimate moments of busy cycle and number served…. Plug in…

22 Example M/M/1/K “like” systems (D. Perry, Boxma, et. al.) Customers that have to wait more than 5 time units will not enter the queue.

23 Infinite Supply Re-entrant Lines

24 Infinite Supply Re-entrant Line

25 “Renewal Like”

26 Thank You

27 Extensions

28 Generator Transitions without events Transitions with events Method: Markovian Arrival Process

29 MMPP (Markov Modulated Poisson Process) Example: rate 4 Poisson Process rate 2 rate 3 rate 4 rate 2 rate 4 rate 3 rate 2 rate 3 rate 4 rate 2 Proposition Transitions without events Transitions with events Fully Counting MAP

30 A Push-Pull Queueing Network

31 2 job streams, 4 steps Queues at 2 and 4 Infinite job supply at 1 and 3 2 servers The Push-Pull Network Control choice based on No idling, FULL UTILIZATION Preemptive resume Push Pull Push Pull

32 Policies Inherently stable Inherently unstable Policy: Pull priority (LBFS) Policy: Linear thresholds Typical Behavior: 2, ,3 Typical Behavior: Server: “don’t let opposite queue go below threshold” Push Pull Push 1,3

33 KSRS

34 M/G/. Pull Priority MG M G Using the Renewal Reward Method: Number served of type 1, during a cycle is 0 w.p..

35 Network View of the Model or

36 Stability Result QueueResidual is strong Markov with state space Theorem: X(t) is positive Harris recurrent. Proof follows framework of Jim Dai (1995) 2 Things to Prove: 1.Stability of fluid limit model 2.Compact sets are petite Positive Harris Recurrence:

37 Diffusion Scaling Now find a limiting process, such that.

38 Diffusion Limit Theorem: When network is PHR and follows rates, With. 10 dimensional Brownian motion Proof Outline: Use positive Harris recurrence to show,, simple calculations along with functional CLT for renewal processes yields the result.

39 Inherently stable network Inherently unstable network Unbalanced network Completely balanced network Configuration

40 Calculation of Rates Corollary: Under assumption (A1), w.p. 1, every fluid limit satisfies:. - Time proportion server works on k - Rate of inflow, outflow through k Full utilization: Stability:

41 Memoryless Processing (Kopzon et. al.) Inherently stable Inherently unstable Policy: Pull priority Policy: Generalized thresholds Alternating M/M/1 Busy Periods Results: Explicit steady state: Stability (Foster – Lyapounov) - Diagonal thresholds - Fixed thresholds

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