On a Queue with Marked Compound Poisson Input and Exponentially Distributed Batch Service

Abstract

AbstractIn this paper, we consider the batch arrival with batch service process. We assume that our queueing model has multi-server. Arrivals of customers in batches of various sizes \((1,2, \dots ,k)\) form a marked compound Poisson process; we designate the batches as \(T_{1}, T_{2}, \dots , T_{k}\). The service time of \(T_{i}\) follows exponential distribution with parameter \(\mu _{i}, i = 1,2,\dots , k\); they are served in batches of the specific size. In arrival process, waiting room of type i for \(T_{i}\) has finite capacity except waiting room of type 1 for \(T_{1}\). In service process, server room of type i has finite capacity. \(T_{i}\) can go to service if server room of type i has available space. If server room of type i does not have available place, then there are two following cases. The first case, if waiting room of type i has available places, \(T_{i}\) must wait in this waiting room. The second case, if waiting room of type i does not have available place, then \(T_{i}\) must leave the system without service except \(T_{1}\), who must wait in waiting room of type 1. Various performance measures are estimated with numerical solution.KeywordsMarked Compound Poisson ProcessBatch ArrivalBatch ServiceMatrix Analytic Method