We have divided this review into two parts. The first part is concerned with the optimal design of queueing systems and the second part deals with the optimal control of queueing systems. The second part, which has the lion’s share of the review since it has received the most attention, focuses mainly on the modelling aspects of the problem and describes the different kinds of threshold (control) policy models available in the literature. To limit the scope of this survey, we decided to limit ourselves to research on papers dealing with the three policies (N, T, and D), where a cost function is designed specifically and optimal thresholds that yield minimum cost are sought.

Optimal design and control of queues

Queueing systems with server vacations (server's performing non-queueing jobs) have been studied extensively since the late 70's. A considerable number of works in this area were completed in the early 80's and surveyed by Doshi in 1986. As an extension to the classical queueing system, allowing idle servers to work on non-queueing jobs makes the vacation models more applicable in a variety of systems including flexible manufacturing or service and computer communication systems. Motivated by these applications, more studies on vacation models have been done during the late 80's and 90's and surveyed in the book by Takagi in 1991 and the book by Tian and Zhang in 2006. This paper intends to provide a brief summary of the most recent research works on vacation queueing systems in the past 10 years. Keywords  server's vacation, batch arrivals, modified vacation policy, working vacations, multi-server vacation models.

http://www.orstw.org.tw/ijor/vol7no4/2-Vol.%207,%20No.4%20pp.3-8.pdf

Recent Developments in Vacation Queueing Models : A Short Survey

An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair

A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation

The N-policy for an unreliable server with delaying repair and two phases of service

Steady state analysis of an Mx/G/1 queue with two phase service and Bernoulli vacation schedule under multiple vacation policy

We consider an M X /G/1 queueing system with a second optional service channel under N-policy. The server remains idle until the queue size reaches or exceeds N (≥1). As soon as the queue size becomes at least N, the server immediately begins to serve the first essential service to all the waiting customers. After the completion of which, only some of them receive the second optional service. For this model, our study is basically concentrated in obtaining the queue size distribution at a random epoch as well as at a departure epoch. Further, we derive a simple procedure to obtain optimal stationary policy under a suitable linear cost structure. Moreover, we provide some important performance measures of this model with some numerical examples.

A Batch Arrival Queue with a Second Optional Service Channel Under N-Policy

This paper deals with an M/G/1 queue with two phases of heterogeneous services and Bernoulli feedback system, where the server provides first phase of regular service to all the customers. As soon as the first phase of service of a customer is completed, it may leave the system or may immediately go for second phase of optional service in one additional service channel. However, after receiving the first phase or second phase of unsuccessful service by an unit, then it may immediately join the tail of the original queue as feedback customer to have another regular service. In this paper, first we derive the queue size distribution at random epoch and at a service completion epoch. Secondly, we derive the distribution of response time and busy period. Finally, we present some numerical results for mean response time.

Madhuchanda Paul

Papers

Steady state analysis of an Mx/G/1 queue with two phase service and Bernoulli vacation schedule under multiple vacation policy

A Batch Arrival Queue with a Second Optional Service Channel Under N-Policy

A two phase queueing system with Bernoulli feedback

A batch arrival queue with an additional service channel under N-policy

A two phases queueing system with Bernoulli vacation schedule under multiple vacation policy