The BMAP/GI/1 queue with server set-up times and server vacations

1993 ◽  
Vol 25 (1) ◽  
pp. 235-254 ◽  
Author(s):  
Josep M. Ferrandiz
Keyword(s):  
Characteristic Function ◽  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Moment Generating Function ◽  
Server Vacations ◽  
Service Completion ◽  
Set Up ◽  
Vacation Period ◽  
Completion Times

Using Palm-martingale calculus, we derive the workload characteristic function and queue length moment generating function for the BMAP/GI/1 queue with server vacations. In the queueing system under study, the server may start a vacation at the completion of a service or at the arrival of a customer finding an empty system. In the latter case we will talk of a server set-up time. The distribution of a set-up time or of a vacation period after a departure leaving a non-empty system behind is conditionally independent of the queue length and workload. Furthermore, the distribution of the server set-up times may be different from the distribution of vacations at service completion times. The results are particularized to the M/GI/1 queue and to the BMAP/GI/1 queue (without vacations).

The BMAP/GI/1 queue with server set-up times and server vacations

1993 ◽  
Vol 25 (01) ◽  
pp. 235-254 ◽  
Author(s):  
Josep M. Ferrandiz
Keyword(s):  
Characteristic Function ◽  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Moment Generating Function ◽  
Server Vacations ◽  
Service Completion ◽  
Set Up ◽  
Vacation Period ◽  
Completion Times

Using Palm-martingale calculus, we derive the workload characteristic function and queue length moment generating function for the BMAP/GI/1 queue with server vacations. In the queueing system under study, the server may start a vacation at the completion of a service or at the arrival of a customer finding an empty system. In the latter case we will talk of a server set-up time. The distribution of a set-up time or of a vacation period after a departure leaving a non-empty system behind is conditionally independent of the queue length and workload. Furthermore, the distribution of the server set-up times may be different from the distribution of vacations at service completion times. The results are particularized to the M/GI/1 queue and to the BMAP/GI/1 queue (without vacations).

Relationships and decomposition in the delayed bernoulli feedback queueing system

1988 ◽  
Vol 25 (1) ◽  
pp. 169-183 ◽  
Author(s):  
D. König ◽  
M. Miyazawa
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Arrival Process ◽  
Stieltjes Transform ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Workload Distribution ◽  
Feedback Queue

For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

scholarly journals Supplementary variable method applied to the MAP/G/1 queueing system

1998 ◽  
Vol 40 (1) ◽  
pp. 86-96 ◽  
Author(s):  
Bong Dae Choi ◽  
Gang Uk Hwang ◽  
Dong Hwan Han
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Service Time ◽  
Queue Length ◽  
Queueing System ◽  
Variable Method ◽  
Supplementary Variable ◽  
Supplementary Variable Method ◽  
One Dimensional

AbstractIn this paper we consider the MAP/G/1 queueing system with infinite capacity. In analysis, we use the supplementary variable method to derive the double transform of the queue length and the remaining service time of the customer in service (if any) in the steady state. As will be shown in this paper, our method is very simple and elegant. As a one-dimensional marginal transform of the double transform, we obtain the generating function of the queue length in the system for the MAP/G/1 queue, which is consistent with the known result.

On a non-Markovian time-dependent queueing system

1989 ◽  
Vol 26 (1) ◽  
pp. 142-151 ◽  
Author(s):  
S. D. Sharma
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Transient State ◽  
Time Dependent ◽  
Queue Length Distribution ◽  
State Behaviour

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.

scholarly journals Discrete-Time Bulk Queueing System with Variable Service Capacity Depending on Previous Service Time

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Yutae Lee
Keyword(s):  
Generating Function ◽  
Discrete Time ◽  
Service Time ◽  
Queue Length ◽  
Queueing System ◽  
Function Technique ◽  
Supplementary Variable ◽  
Service Capacity ◽  
Bulk Service ◽  
Bulk Arrival

This paper considers a discrete-time bulk-arrival bulk-service queueing system with variable service capacity, where the service capacity varies depending on the previous service time. Using the supplementary variable method and the generating function technique, we obtain the queue length distributions at arbitrary slot boundaries and service completion epochs.

On a non-Markovian time-dependent queueing system

1989 ◽  
Vol 26 (01) ◽  
pp. 142-151
Author(s):  
S. D. Sharma
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Transient State ◽  
Time Dependent ◽  
Queue Length Distribution ◽  
State Behaviour

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.

Correlation functions in the G/M/1 system

1980 ◽  
Vol 12 (2) ◽  
pp. 530-540 ◽  
Author(s):  
S. K. Srinivasan ◽  
C. Chudalaimuthu Pillai
Keyword(s):  
Correlation Functions ◽  
Queue Length ◽  
Recurrence Relations ◽  
Queueing System ◽  
Point Method ◽  
Correlation Structure ◽  
Set Up ◽  
Regeneration Point

The G/G/1 queueing system is studied by means of the regeneration point method, exploiting the concept of busy cycles. Recurrence relations are set-up for the distribution of the queue length at the arrival epochs. The same method is used to obtain the correlation structure of arrivals and departures for the G/M/1 queue.

Relationships and decomposition in the delayed bernoulli feedback queueing system

1988 ◽  
Vol 25 (01) ◽  
pp. 169-183
Author(s):  
D. König ◽  
M. Miyazawa
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Arrival Process ◽  
Stieltjes Transform ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Workload Distribution ◽  
Feedback Queue

For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

Correlation functions in the G/M/1 system

1980 ◽  
Vol 12 (02) ◽  
pp. 530-540 ◽  
Author(s):  
S. K. Srinivasan ◽  
C. Chudalaimuthu Pillai
Keyword(s):  
Correlation Functions ◽  
Queue Length ◽  
Recurrence Relations ◽  
Queueing System ◽  
Point Method ◽  
Correlation Structure ◽  
Set Up ◽  
Regeneration Point

The G/G/1 queueing system is studied by means of the regeneration point method, exploiting the concept of busy cycles. Recurrence relations are set-up for the distribution of the queue length at the arrival epochs. The same method is used to obtain the correlation structure of arrivals and departures for the G/M/1 queue.

Sign in / Sign up

Export Citation Format

Share Document