Abstract
In this paper we consider a finite capacity queuing system in which arrivals are governed by a Markovian arrival process. The system is attended by two exponential servers, who offer services in groups of varying sizes. The service rates may depend on the number of customers in service. Using Markov theory, we study this finite capacity queuing model in detail by obtaining numerically stable expressions for (a) the steady‐state queue length densities at arrivals and at arbitrary time points; (b) the Laplace‐Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. The stationary waiting time distribution is shown to be of phase type when the interarrival times are of phase type. Efficient algorithmic procedures for computing the steady‐state queue length densities and other system performance measures are discussed. A conjecture on the nature of the mean waiting time is proposed. Some illustrative numerical examples are presented.
Highlights
Chakravarthy, Srinivas and Alfa, Attahiru S., "A Finite Capacity Queue with Markovian Arrivals and Two Servers with Group Services" (1994)
In this paper we consider a finite capacity queuing system in which arrivals are governed by a Markovian arrival process
Using Markov theory, we study this finite capacity queuing model in detail by obtaining numerically stable expressions for (a) the steady-state queue length densities at arrivals and at arbitrary time points; (b) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals
Summary
We consider a finite capacity queuing system in which arrivals occur singly according to a Markovian arrival process (MAP). In all the above applications, we see that the jobs that require processing of general type can be processed in groups of varying sizes, which motivates the need for the type of service mechanism considered in this paper. A number of optimization problems, useful in the design of such queuing systems, can be studied in terms of choosing a value for L, by fixing the parameters of the arrival and service processes. One such example would be to choose a value of L for which the jobs do not have to wait for a long time before entering service. A conjecture, based on our computational experience, on the nature of the mean waiting time at an arrival epoch is proposed
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