Optimal Thresholds of an Infinite Buffer Discrete-Time Two-Server System with Triadic Policy

Introduction

Discrete-time queuing models have received considerable growing interest during the last few decades due to their potential applications to a variety of slotted digital computer, communication systems. These queuing models are more accurate and efficient than their continuous-time counterparts to analyze and design digital transmitting systems. Extensive analysis of a wide variety of discrete-time queuing models have been reported in Bruneel and Kim (1993), Chaudhry (2000), Gravey and Hbuterne (1992), Hunter (1983), Takagi (1993), and Woodward (1994).

The queuing systems with control (threshold) policy in which the number of working servers can be adjusted one at a time at any arrival epoch or at any service completion epoch depending on the number of customers have been discussed by many researchers. Surveys of single-threshold continuous-time queues have been reported in Tadj and Choudhury (2005). The batch arrival M/G/1 queueing system with Bernoulli vacation schedule and two phases of service under an N policy have been discussed in Choudhury and Madan (2005). Choudhury and Paul (2006) studied an N policy of model with second optional channel for the batch arrival queue. An M/G/1 queueing system with server breakdowns and startup under NT policy was investigated by Ke (2006). Ke (2008) examined system characteristics and optimal two-threshold policy of a batch arrival M/G/1 queueing system with a modified T policy. Ke and Chu (2008) derived the explicit distributions of the randomized T and N policies for the M/G/1 queueing system. Brouns and Van Der Wal (2006) have analyzed optimal threshold policies in a two-class preemptive priority queue. Choudhury and Deka (2008) discussed an M/G/1 retrial queueing system with two phases of service subject to the server breakdown and repair. An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair has been reported in Choudhury and Tadj (2009). Some related works have been reported in Wang (1995, 2003), Wang et al. (2004), Efrosinin (2008), and Tadj and Choudhury (2009). Cost framework for evaluation of information technology has been discussed in Pathak and Vidyarthi (2011).

Lin and Ke (2009) discussed the optimal operating policy for a controllable queuing model in which cost elements, arrival rate and service rate are all fuzzy numbers. Yang et al. (2008) analyzed an analytical optimization analysis of a randomized T policy for an unreliable server M/G/1 queuing system with second optional service. The busy period distribution of an infinite capacity M/M/2 queuing system operating under triadic policy has been discussed in Rhee and Sivazlian (1990). Wang and Wang (2002) studied the optimization of finite capacity queuing system under triadic policy.

It has important applications in the design of control policies for high-performance computer and telecommunication systems, as well as manufacturing systems. With these two service rate control, the service resources or servers’ times can be better allocated to both primary and secondary uses to improve the operational economy of a multiserver queuing system. Fast food restaurant or supermarket cashiers, bank tellers, and telephone service operators are usually multi-task employees. Some of them might perform other secondary jobs when they are not busy and come back to serve waiting customers again when the queue becomes long again. This study is motivated by the popularity of these practical multi-task-server systems and the scarcity of the research work in this area.