The asymptotic variance of departures in critically loaded queues

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies lim t→∞varD(t) / t = λ(1–2/π)(c a 2 + c s 2), where λ is the arrival rate, and c a 2 and c s 2 are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance of D(t) for any t.

[1]  A. P. Zwart,et al.  Tail Asymptotics for the Busy Period in the GI/G/1 Queue , 2001, Math. Oper. Res..

[2]  N. U. Prabhu,et al.  Stochastic Storage Processes , 1980 .

[3]  D. Iglehart,et al.  Multiple channel queues in heavy traffic. I , 1970, Advances in Applied Probability.

[4]  D. König,et al.  Queueing Networks: A Survey of Their Random Processes , 1985 .

[5]  P. Burke.  The Output of a Queuing System , 1956 .

[6]  A. Wald.  On Cumulative Sums of Random Variables , 1944 .

[7]  Zhang Hanqin,et al.  MULTIPLE CHANNEL QUEUES IN HEAVY TRAFFIC , 1990 .

[8]  Ward Whitt.  Complements to heavy traffic limit theorems for the GI/G/ 1 queue , 1972 .

[9]  G. J. Miltenburg,et al.  Variance of the number of units produced on a transfer line with buffer inventories during a period of lengthT , 1987 .

[10]  Kevin B. Hendricks,et al.  The Output Processes of Serial Production Lines of Exponential Machines with Finite Buffers , 1992, Oper. Res..

[11]  Michel Mandjes,et al.  Large Deviations for Gaussian Queues , 2007 .

[12]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[13]  Daryl J. Daley,et al.  Further second-order properties of certain single-server queueing systems , 1975 .

[14]  Michel Mandjes,et al.  Large Deviations for Gaussian Queues: Modelling Communication Networks , 2007 .

[15]  Ward Whitt,et al.  Comparison methods for queues and other stochastic models , 1986 .

[16]  Gerold Alsmeyer,et al.  On Generalized Renewal Measures and Certain First Passage Times , 1992 .

[17]  D. Iglehart,et al.  Multiple channel queues in heavy traffic. II: sequences, networks, and batches , 1970, Advances in Applied Probability.

[18]  Yoni Nazarathy,et al.  The variance of departure processes: puzzling behavior and open problems , 2011, Queueing Syst. Theory Appl..

[19]  N. U. Prabhu,et al.  Stochastic Storage Processes: Queues, Insurance Risk, Dams, and Data Communication , 1997 .

[20]  P. Kuehn,et al.  Approximate Analysis of General Queuing Networks by Decomposition , 1979, IEEE Trans. Commun..

[21]  W. Whitt,et al.  The Queueing Network Analyzer , 1983, The Bell System Technical Journal.

[22]  Ralph L. Disney,et al.  Traffic processes in queueing networks: a Markov renewal approach , 1987 .

[23]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[24]  Vincent Hodgson,et al.  The Single Server Queue. , 1972 .

[25]  R. Wolff.  AN UPPER BOUND FOR MULTI-CHANNEL QUEUES , 1977 .

[26]  Yoni Nazarathy,et al.  The asymptotic variance rate of the output process of finite capacity birth-death queues , 2008, Queueing Syst. Theory Appl..

[27]  D. Daley,et al.  Queueing output processes , 1976, Advances in Applied Probability.

[28]  John O. McClain,et al.  The output processes of serial production lines of general machines with finite buffers , 1993 .

[29]  Ward Whitt,et al.  Departures from a Queue with Many Busy Servers , 1984, Math. Oper. Res..

[30]  Peter C. Kiessler,et al.  Traffic Processes in Queueing Networks , 1987 .

[31]  A. Gut.  Stopped Random Walks , 1987 .

[32]  Ward Whitt,et al.  Variability functions for parametric-decomposition approximations of queueing networks , 1995 .

[33]  N. L. Lawrie,et al.  Comparison Methods for Queues and Other Stochastic Models , 1984 .

[34]  T. Kamae,et al.  Stochastic Inequalities on Partially Ordered Spaces , 1977 .

[35]  Baris Tan,et al.  Asymptotic variance rate of the output in production lines with finite buffers , 2000, Ann. Oper. Res..

[36]  W. Whitt.  Comparing counting processes and queues , 1981, Advances in Applied Probability.

[37]  Ahmad Al Hanbali,et al.  A Tandem Queueing Model for Delay Analysis in Disconnected Ad Hoc Networks , 2008, ASMTA.