Abstract
This paper deals with a batch arrival infinite-buffer single server queue. The interbatch arrival times are generally distributed and arrivals are occurring in batches of random size. The service process is correlated and its structure is presented through a continuous-time Markovian service process (C-MSP). We obtain the probability density function (p.d.f.) of actual waiting time for the first and an arbitrary customer of an arrival batch. The proposed analysis is based on the roots of the characteristic equations involved in the Laplace-Stieltjes transform (LST) of waiting times in the system for the first, an arbitrary, and the last customer of an arrival batch. The corresponding mean sojourn times in the system may be obtained using these probability density functions or the above LSTs. Numerical results for some variants of the interbatch arrival distribution (Pareto and phase-type) have been presented to show the influence of model parameters on the waiting-time distribution. Finally, a simple computational procedure (through solving a set of simultaneous linear equations) is proposed to obtain the “R” matrix of the corresponding GI/M/1-type Markov chain embedded at a prearrival epoch of a batch.
Highlights
In recent years, analysis of queueing processes with nonrenewal arrival and service processes has been carried out to model data transmission of complex computer and communication networks
Since various distributions can be either represented or approximated by PH-distribution, we take interbatch arrival time distribution to be of PH-type having the representation (α, T), where α and T are of dimension ]
We have obtained component-wise waiting-time distribution function of an infinite-buffer GI/continuous-time Markovian service process (C-MSP)/1 queueing model with parameters given in Table 2 of [12]
Summary
Analysis of queueing processes with nonrenewal arrival and service processes has been carried out to model data transmission of complex computer and communication networks. Once the roots of these characteristic equations are found, it becomes easy to obtain the stationary waiting-time distribution for the first and an arbitrary customer of an arrival batch. A simple computational procedure based on the solution of a set of linear simultaneous equations has been proposed to compute the R matrix of the corresponding GI/M/1-type Markov chain embedded at a prearrival epoch of a batch. It is well-known that the determination of the “R” matrix can be done through the minimal nonnegative solution of a nonlinear matrix equation; see Neuts [10] for details. The present work is a nontrivial extension of [5]
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