Abstract
M/G/1 queues with server vacations have been studied extensively over the last two decades. Recent surveys by Boxma [3], Doshi [5] and Teghem [14] provide extensive summary of literature on this subject. More recently, Shanthikumar [11] has generalized some of the results toM/G/1 type queues in which the arrival pattern during the vacations may be different from that during the time the server is actually working. In particular, the queue length at the departure epoch is shown to decompose into two independent random variables, one of which is the queue length at the departure epoch (arrival epoch, steady state) in the correspondingM/G/1 queue without vacations. Such generalizations are important in the analysis of situations involving reneging, balking and finite buffer cyclic server queues. In this paper we consider models similar to the one in Shanthikumar [11] but use the work in the system as the starting point of our investigation. We analyze the busy and idle periods separately and get conditional distributions of work in the system, queue length and, in some cases, waiting time. We then remove the conditioning to get the steady state distributions. Besides deriving the new steady state results and conditional waiting time and queue length distributions, we demonstrate that the results of Boxma and Groenendijk [2] follow as special cases. We also provide an alternative approach to deriving Shanthikumar's [11] results for queue length at departure epochs.
Similar content being viewed by others
References
P.H. Brill and M.J.M. Posner, Level crossing in point processes applied to queues: single server case, Oper. Res. 25 (1977) 662.
O.J. Boxma and W.P. Groenendijk, Pseudoconservation laws in cyclic-service systems, J. Appl. Prob. 24 (1987) 949.
O.J. Boxma, Workloads and waiting times in single-server systems with multiple customer classes, to appear in Queueing Systems.
R.B. Cooper, Queues served in cyclic order: waiting times, BSTJ 49 (1970) 399.
B.T. Doshi, Queueing systems with vacations — A survey, Queueing Systems 1 (1986) 29.
A. Federgruen and L. Green, Queueing systems with service interruptions, Oper. Res.
S. Fuhrmann and R.B. Cooper, Stochastic decomposition in anM/G/1 queue with generalized vacations, Oper. Res. 33 (1985) 1117.
D.P. Gaver, A waiting line with interrupted service, including priorities, J. Roy. Stat. Soc. B24 (1962) 73.
C.M. Harris and W.G. Marchal, State dependence inM/G/1 server-vacation models, Oper. Res. 36 (1988) 560.
T.J. Ott, On theM/G/1 queue with additional inputs, J. Appl. Prob. 21 (1984) 129.
J.G. Shanthikumar, On stochastic decomposition inM/G/1 type queues with generalized server vacations, Oper. Res. 36 (1988) 566.
J.G. Shanthikumar, Some analyses on the control of queues using level crossings of regenerative processes, J. Appl. Prob. 17 (1980) 814.
J.G. Shanthikumar, Optimal control of anM/G/1 priority queue viaN-control, Am. J. Math. Management Sci. 1 (1981).
J.G. Shanthikumar, Analysis of a single server queue with time and operation dependent server failures, Adv. Management Sci. 1 (1982) 339.
J.G. Shanthikumar and M.J. Chandra, Applications of level crossing analysis to discrete state processes in queueing systems, Naval Res. Logistics Quarterly 29 (1982) 593.
B. Sengupta, A queue with service interruptions in an alternating random environment, to appear in Oper. Res.
H. Takagi,Analysis of Polling Systems (MIT Press, Cambridge, MA, 1986).
J. Teghem, Jr., Control of the service process in a queueing system, Eur. J. Oper. Res. 23 (1986) 141.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Doshi, B. Conditional and unconditional distributions forM/G/1 type queues with server vacations. Queueing Syst 7, 229–251 (1990). https://doi.org/10.1007/BF01154544
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01154544