The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

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

Single server retrial queues with two way communication

SummaryThis paper deals with the stochastic decomposition property for retrial queues. This property is connected with similar results for vacation models. As applications, the moments of the number of customers in orbit and the rate of convergence under high retrial intensity can be obtained.

Stochastic decomposition for retrial queues

(1962). The Theory of Storage. Journal of the Operational Research Society: Vol. 13, No. 1, pp. 104-106.

https://link.springer.com/content/pdf/10.1057%2Fjors.1962.14.pdf

The Theory of Storage

On multiserver feedback retrial queue with finite buffer

In this paper, we are concerned with several random processes that occur in M/M/1 feedback queues with impatient customers in which the feedback decision process is a Bernoulli process. The stationary distribution of the arrival process has been obtained. Results for particular queues without feedback are deduced.

A single server queue with impatient and feedback customers

In this paper, a steady-state Markovian multi-server retrial queueing system with Bernoulli vacation scheduling service is studied. Using matrix-geometric approach, various interesting and important system performance measures are obtained. Further, the probability descriptors like ideal retrial and vain retrial are provided. Finally, extensive numerical illustrations are presented to indicate the quantifying nature of the approach to obtain solutions to this queueing system.

An M/M/C retrial queueing system with Bernoulli vacations

We consider an M/G/1 feedback queue with optional server vacations based on Bernoulli schedule and a single vacation policy. We assume that the server provides two types of service, type 1 with probability p1 and type 2 with probability p2 with the service times following general distribution and each arriving customer may choose either type of service. However after the completion of each type 1 or type 2 service, the customer can feedback to the tail of the original queue with probability p to repeat the service until it is successful or may depart the system with probability q if service happens to be successful. The feedback customer also has the option to choose either type 1 or type 2 service with probability p1 and p2 respectively. Again at the completion of each type of service, the server can take a vacation with probability θ or may continue to stay in the system with probability 1 − θ. Further, we assume that whenever, the server takes a vacation, it is always a single vacation with exponentially distributed vacation period. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Also the mean queue length and the mean waiting time are computed.

A Single Server M/G/1 Feedback Queue with Two Types of Service Having General Distribution

We study a two phase M/G/1 queueing system with Bernoulli feedback where the server takes multiple vacation. All the Poisson arrivals with mean arrival rate (λ > 0) demand first essential service, whereas only some of them demand second optional service. The service times of the first essential service are assumed to follow a general distribution function B 1(v) and that the second optional service with general distribution with distribution function B 2(v). However after completion of the first service or second service, if the customer is dissatisfied with its service, he can immediately join the tail of the queue as a feedback customer for receiving another regular service with probability p. Otherwise the customer may depart forever from the system with probability q = 1 −p. If there is no customer in the queue, then the server goes for vacation and the vacation periods are exponentially distributed with mean vacation time . On returning from vacation if the server again founds no customer wa...

V. Thangaraj

Papers

On multiserver feedback retrial queue with finite buffer

A single server queue with impatient and feedback customers

An M/M/C retrial queueing system with Bernoulli vacations

A Single Server M/G/1 Feedback Queue with Two Types of Service Having General Distribution

A Two Phase M/G/1 Feedback Queue with Multiple Server Vacation