Analysis of a Dual Tandem Queue with Servers Floating Between the Stages

Abstract

A tandem queueing system with a correlated arrival process and two multi-server stages is analyzed. The capacity of the buffer at stage 1 is infinite. The capacity of the bufferat stage 2 is finite. The total number of available servers is fixed. Servers are dynamically shared between the stages in such a way that any server cannot stay idle if at least one of the buffers is not idle. Servers can transit between the stages only at service completion epochs. Servers from stage 2 can transit to stage 1 even if the buffer at stage 2 is not empty, according to the control policy defined by two integer thresholds. If service by the server assigned to stage 2 is completed when the number of customers in the stage 1 buffer is not less than the first threshold and the number of customers in the stage 2 buffer is less than the second threshold, the released server is re-assigned to stage 1 and immediately starts service. If service by a server assigned to stage 1 is completed when the stage 2 buffer is full, the released server is re-assigned to stage 2 and immediately starts service. In the case of service completion at stage 2 during the epoch when the buffers at both stages are idle, the released server is re-assigned to stage 1 and waits for a new customer arrival at this stage. Customers’ arrival is described by the Markov arrival process (MAP). Each customer has to receive service at both stages of the tandem or only at stage 1. The service times at both stages have a phase-type distribution with parameters depending on the stage. Under the fixed values of the thresholds, analysis of the stationary behavior of the tandem is implemented, including derivation of the ergodicity condition, computation of the stationary distribution of the number of customers at each stage, and derivation of expressions for the key performance indicators. Analysis is essentially based on the proper use of the notion of the generalized phase-type distribution. The results of numerical experiments illustrating the feasibility of the proposed algorithms and highlighting the dependence of the performance measures of the system on the parameters of the control policy are presented. The problem of the optimal choice of thresholds is briefly considered.