On the variance curve of outputs for some queues and networks Yoni Nazarathy Gideon Weiss Yoav Kerner QPA Seminar, EURANDOM January 8, 2009

2 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

3 Outline of the Talk: Models and Methods Models: Finite Capacity Birth Death Queues (M/M/1/K) General Lossless Queues M/G/1 Queue Push-Pull (infinite supply) Network Infinite supply re-entrant line Methods: Markovian Arrival Process (MAP) Embedding in Renewal Reward Regenerative Simulation (Renewal Reward) Diffusion Limits

4 Finite Capacity Birth-Death Queues

5 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)

6 BRAVO Effect (M/M/1/K)

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

8 General Lossless Queues

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

10 M/G/1 Queue

11 M/G/1 Linear Asymptote Theorem:

12 Shape of Variance Curve (?) Pas Op: Possible non-sense ahead!!!

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

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

15 Using in Regenerative Simulation

16 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…

17 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.

18 A Push-Pull Queueing Network

19 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

20 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

21 KSRS

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

23 Using Diffusion Limits Now assume general processing times with finite second moment.

24 Network View of the Model or

25 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:

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

27 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.

28 Infinite Supply Re-entrant Lines

29 Infinite Supply Re-entrant Line

30 “Renewal Like”

31 Thank You

32 Extensions

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

34 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:

35 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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41 Proof Outline Whitt: Book: Stochastic Process Limits,. Paper: Asymptotic Formulas for Markov Processes… 1) Lemma: Look at M(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 of asymptotic variance rate of birth-death MMPP.

42 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

43 01 K K – 1 Some intuition for M/M/1/K-BRAVO …

44 M/M/40/40 M/M/10/10 M/M/1/40 K=20 K=30 c=30 c=20

45 MAP used for PH/PH/1/40 with Erlang and Hyper-Exp distributions

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

47 For Large K

48 Proposition: For, M/M/1/K