The GI/M/1 queue with exponential vacations

“…In the M/PH/1 queue with working vacations, the stationary queue length is distributed as the conditional sum of the stationary queue length in an M/PH/1 queue without vacations and the number of customers in the system at an arbitrary time during the vacation. Note that, the block forms of the MSP are based on two types of phases of the MSP and similar factorizations (decompositions) have been obtained for vacation models in Refs [19,20].…”

Section: Introductionmentioning

confidence: 99%

“…We refer to the service process as a Markovian service process (MSP) and study MAP/MSP/1 queues. In Refs [19,20], the idea of using two types of states (phases) was used in analyzing queueing models with vacations. This idea is extended to a general service discipline in our model.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Queues with Markovian Service Processes

Ozawa

2004

Stochastic Models

View full text Add to dashboard Cite

Markovian service process (MSP) is a model similar to the Markovian arrival process (MAP), where arrivals are replaced with service completions. The MSP can represent various queueing models such as vacation models, N -policy models and exceptional service models. We analyze MAP/MSP/1 queues and obtain a new sort of matrix-type factorization of the vector generating function for the stationary queue length. The MAP/MSP/1 queue is a very tractable model since its behavior is represented as a quasi-birth-and-death process.

show abstract

“…Some literatures on this type of vacation queue have emerged. In 1989, Tian et al [6] applied the matrix analytic method to analyze a / /1 queue with exhaustive service and multiple exponential vacations. Chatterjee and Mukherjee [7] analyzed the same model as that in Tian et al [6].…”

Section: Introductionmentioning

confidence: 99%

“…In 1989, Tian et al [6] applied the matrix analytic method to analyze a / /1 queue with exhaustive service and multiple exponential vacations. Chatterjee and Mukherjee [7] analyzed the same model as that in Tian et al [6]. But in [7], the vacation times are independently and identically distributed with a general distribution.…”

Section: Introductionmentioning

confidence: 99%

On the GI/M/1 Queue with Vacations and Multiple Service Phases

Liu

2017

Mathematical Problems in Engineering

View full text Add to dashboard Cite

This paper considers a / /1 queue with vacations and multiple service phases. Whenever the system becomes empty, the server takes a vacation, causing the system to move to vacation phase 0. If the server returns from a vacation to find no customer waiting, another vacation begins. Otherwise, the system jumps from phase 0 to some service phase with probability , = 1, 2, . . . , . Using the matrix geometric solution method and semi-Markov process, we obtain the distributions of the stationary system size at both arrival and arbitrary epochs. The distribution of the stationary waiting time of an arbitrary customer is also derived. In addition, we present some performance measures such as mean waiting time of an arbitrary customer, mean length of the typecycle, and mean number of customers in the system at the end of phase 0. Finally, some numerical examples are presented.

show abstract

The GI/M/1 queue with exponential vacations

Cited by 48 publications

References 10 publications

G/M^<a, b>/1 QUEUES WITH SERVER VACATIONS

G/M^<a, b>/1 QUEUES WITH SERVER VACATIONS

Analysis of Queues with Markovian Service Processes

On the GI/M/1 Queue with Vacations and Multiple Service Phases

Contact Info

Product

Resources

About