AGI/M/1 queue with a modified service mechanism

118

In this note we prove some stochastic decomposition results for variations of the GI/G/1 queue. Our main model is a GI/G/1 queue in which the server, when it becomes idle, goes on a vacation for a random length of time. On return from vacation, if it finds customers waiting, then it starts serving the first customer in the queue. Otherwise it takes another vacation and so on. Under fairly general conditions the waiting time of an arbitrary customer, in steady state, is distributed as the sum of two independent random variables: one corresponding to the waiting time without vacations and the other to the stationary forward recurrence time of the vacation. This extends the decomposition result of Gelenbe and Iasnogorodski [5]. We use sample path arguments, which are also used to prove stochastic decomposition in a GI/G/1 queue with set-up time.

“…The OJ/0/1/ S model and its special cases have been studied by various authors [10], [12], [13], [17], [19], [23]. However, the decomposition result we obtain here is new.…”

Section: Introductionmentioning

confidence: 61%

“…Successive set-up times {Y n } form a renewal sequence independent of {An} and {Sn}. A related model was studied by Lemoine [7] and Minh [10] (also see [12], [13], [19], [23]). In their model the customer arriving to an idle system has a different service time than the others.…”

Section: Stochastic Decomposition In the Gilgills Queuementioning

confidence: 99%

A note on stochastic decomposition in a GI/G/1 queue with vacations or set-up times

Doshi

1985

118

“…sequence mutually independent of all other underlying random variables. This model has been treated by Welch (1964) in the MIGll context, by Pakes (1972) and Shanthikumar (1982) in the G1M11 context, and by Doshi (1985), Lemoine (1975), Minh (1980) and Pakes (1973) in the GIG/l context. For this model the following theorem holds trivially.…”

Section: Warm-up Timesmentioning

confidence: 99%

Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications

Shanthikumar¹,

Sumita²

1989

We consider a discrete-time stochastic process {Wn, n≧0} governed by i.i.d random variables {ξ n} whose distribution has support on (–∞,∞) and replacement random variables {Rn} whose distributions have support on [0,∞). Given Wn, Wn+ 1 takes the value Wn + ζ n+ 1 if it is non-negative. Otherwise Wn+ 1 takes the value Rn +1 where the distribution of Rn+ 1 depends only on the value of Wn + ζn +1. This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn = 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.

“…This system is called a modified GI IG11 queue. Modified M IG/1 has been investigated by Yeo (1962) and Welch (1964) and modified GI 1M 11 by Pakes (1972).…”

Section: Introductionmentioning

confidence: 99%

On the busy period of the modified GI/G/1 queue

Pakes

1973

Self Cite

Proceeding from duality results for the GI/G/1 queue, this paper obtains the probability of the number served in a busy period of a GI/G/1 system where customers initiating a busy period have a different service time distribution from other customers. Using duality arguments for processes with interchangeable increments, the Laplace transform of the busy period duration is found for a modified GI/M/1 queue.