Abstract
Departure processes in infinite server queues with non-Poisson arrivals have not received much attention in the past. In this paper, we try to fill this gap by considering the continuous time departure process in a general infinite server system with a Markov renewal batch arrival process ofM different types of customers. By a conditional approach, analytical results are obtained for the generating functions and binomial moments of the departure process. Special cases are discussed, showing that while Poisson arrival processes generate Poisson departures, departure processes are much more complicated with non-Poisson arrivals.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
E. Çinlar,Introduction to Stochastic Processes (Prentice-Hall, Englewood Cliffs, NJ, 1975).
D.J. Daley, Queueing departure processes, Adv. Appl. Prob. 8 (1976) 395–415.
R.L. Disney and R.C. Kiessler,Traffic Processes in Queueing Networks, A Markov Renewal Approach (The Johns Hopkins University Press, Baltimore, 1987).
J.L. Doob,Stochastic Processes (Wiley, New York, 1953).
S.G. Eick, W.A. Massey and W. Whitt, TheM t /G/∞ queue, AT&T Bell Lab. (1990).
J.M. Harrison and A.J. Lemoine, A note on networks of infinite-server queues, J. Appl. Prob. 18 (1981) 561–567.
L. Liu, B.R.K. Kashyap and J.G.C. Templeton, On theGI X /G/∞ system, J. Appl. Prob. 27 (1990) 671–683.
L. Liu and J.G.C. Templeton, The\(G^{X{n}} /G_n /\infty \) system: system size, Queueing Systems 8 (1991) 323–356.
L. Liu and J.G.C. Templeton, Autocorrelations in infinite server batch arrival queues, Queueing Systems 14 (1993) 313–337.
W.A. Massey and W. Whitt, Networks of infinite-server queues with non-stationary Poisson input, Queueing Systems 13 (1993) 183–250.
N.M. Mirasol, The output of anM/G/∞ queueing system is Poisson, Oper. Res. 11 (1963) 282–284.
M.F. Ramalhoto, Some statistical problems in random translations of stochastic point processes, Ann. Oper. Res. 8 (1987) 229–242.
J.F. Reynolds, The covariance structure of queues and related processes — A survey of recent work, Adv. Appl. Prob. 7 (1975) 383–415.
V. Schmidt, On the joint queue-length characteristics in infinite-server tandem queues with heavy traffic, Adv. Appl. Prob. 19 (1987) 474–486.
D.N. Shanbhag, On infinite server queues with batch arrivals, J. Appl. Prob. 3 (1966) 274–279.
D.A. Stanford, Interdeparture-time distributions in the non-preemptive priority σMi/Gi/1 queue, Perform. Eval. 12 (1991) 43–60.
W. Whitt, Departures from a queue with many busy servers, Math. Oper. Res. 9 (1984) 534–544.
W. Whitt, Approximations for departure processes and queues in series, Naval Res. Logist. Quart. 31 (1984) 499–521.
Author information
Authors and Affiliations
Additional information
This research has been supported in part by the Natural Science and Engineering Research Council of Canada (Grant A5639).
Rights and permissions
About this article
Cite this article
Liu, L., Templeton, J.G.C. Departures in\(GR^{X{n}} /G_n /\infty \) . Queueing Syst 19, 399–419 (1995). https://doi.org/10.1007/BF01151931
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01151931