Analysis of Finite Buffer Markovian Queue with Balking, Reneging and Working Vacations

2013 ◽  
Vol 4 (1) ◽  
pp. 1-24 ◽  
Author(s):  
P. Vijaya Laxmi ◽  
V. Goswami ◽  
K. Jyothsna
Keyword(s):  
Performance Measures ◽  
Finite Buffer ◽  
State Behavior ◽  
Arrival Times ◽  
Service Period ◽  
Markovian Queue ◽  
Special Cases ◽  
Working Vacations ◽  
Service Times ◽  
Vacation Period

This article presents the analysis of a finite buffer M/M/1 queue with multiple and single working vacations. The arriving customers balk (that is do not join the queue) with a probability and renege (that is leave the queue after joining) according to exponential distribution. The inter-arrival times, service times during a regular service period, service times during a vacation period and vacation times are independent and exponentially distributed random variables. Steady-state behavior of the model is considered and various performance measures, some special cases of the model and cost analysis are discussed.

DISCRETE-TIME GIX/GEO/1/N QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTION

2014 ◽  
Vol 31 (01) ◽  
pp. 1450003 ◽  
Author(s):  
SHAN GAO ◽  
ZAIMING LIU ◽  
QIWEN DU
Keyword(s):  
Discrete Time ◽  
Finite Buffer ◽  
Waiting Time Distribution ◽  
Service Period ◽  
Imbedded Markov Chain ◽  
Working Vacations ◽  
Vacation Interruption ◽  
Service Times ◽  
System Length ◽  
Vacation Period

In this paper, we study a discrete-time finite buffer batch arrival queue with multiple geometric working vacations and vacation interruption: the server serves the customers at the lower rate rather than completely stopping during the vacation period and can come back to the normal working level once there are customers after a service completion during the vacation period, i.e., a vacation interruption happens. The service times during a service period, service times during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer's observation epochs. We also present probability generation function (p.g.f.) of actual waiting-time distribution in the system and some performance measures.

scholarly journals Finite Buffer GI/M(n)/1 Queue with Bernoulli-Schedule Vacation Interruption under N-Policy

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
P. Vijaya Laxmi ◽  
V. Suchitra
Keyword(s):  
Performance Measures ◽  
Finite Buffer ◽  
Supplementary Variable ◽  
Working Vacation ◽  
Variable Technique ◽  
Special Cases ◽  
Vacation Interruption ◽  
System Length ◽  
Vacation Period ◽  
Bernoulli Schedule

We study a finite buffer N-policy GI/M(n)/1 queue with Bernoulli-schedule vacation interruption. The server works with a slower rate during vacation period. At a service completion epoch during working vacation, if there are at least N customers present in the queue, the server interrupts vacation and otherwise continues the vacation. Using the supplementary variable technique and recursive method, we obtain the steady state system length distributions at prearrival and arbitrary epochs. Some special cases of the model, various performance measures, and cost analysis are discussed. Finally, parameter effect on the performance measures of the model is presented through numerical computations.

scholarly journals A Discrete-Time Queue with Balking, Reneging, and Working Vacations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Veena Goswami
Keyword(s):  
Discrete Time ◽  
Cost Model ◽  
Single Server ◽  
Finite Buffer ◽  
Multiple Working Vacations ◽  
Optimal Service ◽  
Special Cases ◽  
Service Rates ◽  
Working Vacations ◽  
Vacation Period

This paper presents an analysis of balking and reneging in finite-buffer discrete-time single server queue with single and multiple working vacations. An arriving customer may balk with a probability or renege after joining according to a geometric distribution. The server works with different service rates rather than completely stopping the service during a vacation period. The service times during a busy period, vacation period, and vacation times are assumed to be geometrically distributed. We find the explicit expressions for the stationary state probabilities. Various system performance measures and a cost model to determine the optimal service rates are presented. Moreover, some queueing models presented in the literature are derived as special cases of our model. Finally, the influence of various parameters on the performance characteristics is shown numerically.

scholarly journals NOTE ON APPROXIMATIONS FOR THE MULTISERVER QUEUE WITH FINITE BUFFER AND DETERMINISTIC SERVICES

2008 ◽  
Vol 22 (4) ◽  
pp. 653-658 ◽  
Author(s):  
Henk Tijms
Keyword(s):  
Performance Measures ◽  
Service Time ◽  
Finite Buffer ◽  
Two Phase ◽  
Sojourn Times ◽  
Deterministic Service ◽  
Service Times ◽  
Phase Process

This article shows that very accurate accurate approximations to performance measures in the multiserver M/D/c/c+N queue with finite buffer and deterministic service times can be obtained by replacing the deterministic service time by a two-phase process with exponential sojourn times and branching probabilities outside the interval [0, 1].

scholarly journals Optimization of Balking and Reneging Queue with Vacation Interruption underN-Policy

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
P. Vijaya Laxmi ◽  
V. Goswami ◽  
K. Jyothsna
Keyword(s):  
Performance Measures ◽  
Search Method ◽  
Model Parameters ◽  
Finite Buffer ◽  
Working Vacation ◽  
Swarm Optimization ◽  
Multiple Working Vacations ◽  
Working Vacations ◽  
Vacation Interruption ◽  
System Length

This paper analyzes a finite buffer multiple working vacations queue with balking, reneging, and vacation interruption underN-policy. In the working vacation, a customer is served at a lower rate and at the instants of a service completion; if there are at leastNcustomers in the queue, the vacation is interrupted and the server switches to regular busy period otherwise continues the vacation. Using Markov process and recursive technique, we derive the stationary system length distributions at arbitrary epoch. Various performance measures and some special models of the system are presented. Cost analysis is carried out using particle swarm optimization and quadratic fit search method. Finally, some numerical results showing the effect of model parameters on key performance measures of the system are presented.

scholarly journals Analysis of Impatient Customers in Queues with Bernoulli Schedule Working Vacations and Vacation Interruption

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Veena Goswami
Keyword(s):  
Performance Measures ◽  
System Size ◽  
Stochastic Decomposition ◽  
Model Parameters ◽  
The Mean ◽  
Working Vacations ◽  
Vacation Interruption ◽  
The Impact ◽  
Vacation Period ◽  
Bernoulli Schedule

This paper analyzes customers’ impatience in Markovian queueing system with multiple working vacations and Bernoulli schedule vacation interruption, where customers’ impatience is due to the servers’ vacation. During the working vacation period, if there are customers in the queue, the vacation can be interrupted at a service completion instant and the server begins a regular busy period with probability 1-q or continues the vacation with probability q. We obtain the probability generating functions of the stationary state probabilities and deduce the explicit expressions of the system sizes when the server is in a normal service period and in a Bernoulli schedule vacation interruption, respectively. Various performance measures such as the mean system size, the proportion of customers served, the rate of abandonment due to impatience, and the mean sojourn time of a customer served are derived. We obtain the stochastic decomposition structures of the queue length and waiting time. Finally, some numerical results to show the impact of model parameters on performance measures of the system are presented.

The analysis of a discrete time finite-buffer queue with working vacations under Markovian arrival process and PH-service time

2020 ◽  
Vol 54 (3) ◽  
pp. 675-691
Author(s):  
Qingqing Ye ◽  
Liwei Liu ◽  
Tao Jiang ◽  
Baoxian Chang
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Loss Probability ◽  
Combination Method ◽  
Arrival Process ◽  
Finite Buffer ◽  
The Matrix ◽  
Working Vacations ◽  
The Impact ◽  
Number Of Customers

In this paper, we study the discrete-time MAP/PH/1 queue with multiple working vacations and finite buffer N. Using the Matrix-Geometric Combination method, we obtain the stationary probability vectors of this model, which can be expressed as a linear combination of two matrix-geometric vectors. Furthermore, we obtain some performance measures including the loss probability and give the limit of loss probability as finite buffer N goes to infinite. Waiting time distribution is derived by using the absorbing Markov chain. Moreover, we obtain the number of customers served in the busy period. At last, some numerical examples are presented to verify the results we obtained and show the impact of parameter N on performance measures.

scholarly journals Analysis of multi server Markovian queue with functioning vacation and intolerance of customer

2018 ◽  
Vol 7 (2.21) ◽  
pp. 448
Author(s):  
S Shanmugasundaram ◽  
R Murugesan
Keyword(s):  
Performance Measures ◽  
Busy Period ◽  
Service Rate ◽  
State Behavior ◽  
Repair Work ◽  
Infinite Population ◽  
Markovian Queue ◽  
Server Failure ◽  
Multi Server ◽  
Steady State Behavior

In this article, we analyze the operating behavior of two server Markovian queueing model with functioning vacation and infinite population. If the server is halt his service suddenly in a normal busy period and repair work is done immediately and service starts. The server failure and repair rates are follow exponential distribution, when the system become vacation the server takes functioning during this period the customer wait in the queue and server serves the customer with the lower service rate. The steady state behavior is also obtained, the various performance measures are also determined. The numerical example is given to test the feasibility of the model.    

scholarly journals On a Retrial Queueing Model with Customer Induced Interruption

2020 ◽  
pp. 112-120
Author(s):  
Varghese Jacob
Keyword(s):  
Poisson Process ◽  
Exponential Distribution ◽  
Performance Measures ◽  
Queueing System ◽  
Single Server ◽  
Finite Buffer ◽  
Preemptive Priority ◽  
State Behavior ◽  
Retrial Queueing System ◽  
Retrial Queueing

This paper presents a retrial queueing system with customer induced interruption while in service. We consider a single server queueing system of infinite capacity to which customers arrive according to a Poisson process and the service time follows an exponential distribution.An arriving customer to an idle server obtains service immediately and customers who find server busy go directly to the orbit from where he retry for service. The inter-retrial time follows exponential distribution. The customer interruption while in service occurs according to a Poisson process and the interruption duration follows an exponential distribution. The customer whose service is got interrupted will enter into a finite buffer. Any interrupted customer, finding the buffer full, is considered lost. Those interrupted customers who complete their interruptions will be placed into another buffer of same size. The interrupted customers waiting for service are given non-preemptive priority over new customers. We analyse the steady-state behavior of this queuing system. Several performance measures are obtained. Numerical illustrations of the system behaviour are also provided with example.

A Note on Bounds and Error Bounds for Nonexponential Batch Arrival Systems

1997 ◽  
Vol 11 (2) ◽  
pp. 189-201 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Nico M. van Dijk
Keyword(s):  
Performance Measures ◽  
Error Bounds ◽  
Sample Path ◽  
Practical Interest ◽  
Batch Arrival ◽  
Finite Buffer ◽  
Infinite Space ◽  
Proof Techniques ◽  
Service Times ◽  
Markov Reward

This note studies the comparison of finite-buffer and nonexponential batch arrival systems of the form Gx/M/c/c + N with the corresponding systems, with N replaced by N', where N' can be smaller, larger, or infinite. If N' = ∞ the service times can be arbitrarily distributed. Both comparison and error bounds are obtained for performance measures such as the throughput, the idle probability, and the active server distribution. The results are of practical interest to establish computational reductions, either by infinite-space approximation or by reduced finite truncations. Two different proof techniques will be employed: the sample path approach and the Markov reward approach. The comparison of these two techniques is of interest in itself, showing the advantage and disadvantage of each.

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