Discrete-time single-server finite-buffer queues under discrete Markovian arrival process with vacations

2007 ◽  
Vol 64 (1) ◽  
pp. 1-19 ◽  
Author(s):  
U.C. Gupta ◽  
S.K. Samanta ◽  
R.K. Sharma ◽  
M.L. Chaudhry
Keyword(s):  
Discrete Time ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Single Server ◽  
Finite Buffer ◽  
Discrete Markovian Arrival Process

scholarly journals A Queueing System with Heterogeneous Impatient Customers and Consumable Additional Items

2017 ◽  
Vol 27 (2) ◽  
pp. 367-384 ◽  
Author(s):  
Janghyun Baek ◽  
Olga Dudina ◽  
Chesoong Kim
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Optimization Problems ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Single Server ◽  
Finite Buffer ◽  
Threshold Strategy

Abstract A single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.

scholarly journals Some performance measures for vacation models with a batch Markovian arrival process

1994 ◽  
Vol 7 (2) ◽  
pp. 111-124 ◽  
Author(s):  
Sadrac K. Matendo
Keyword(s):  
Performance Measures ◽  
Waiting Times ◽  
Distribution Functions ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Service Discipline ◽  
Renewal Processes ◽  
Single Server ◽  
Batch Markovian Arrival Process ◽  
Interarrival Times

We consider a single server infinite capacity queueing system, where the arrival process is a batch Markovian arrival process (BMAP). Particular BMAPs are the batch Poisson arrival process, the Markovian arrival process (MAP), many batch arrival processes with correlated interarrival times and batch sizes, and superpositions of these processes. We note that the MAP includes phase-type (PH) renewal processes and non-renewal processes such as the Markov modulated Poisson process (MMPP).The server applies Kella's vacation scheme, i.e., a vacation policy where the decision of whether to take a new vacation or not, when the system is empty, depends on the number of vacations already taken in the current inactive phase. This exhaustive service discipline includes the single vacation T-policy, T(SV), and the multiple vacation T-policy, T(MV). The service times are i.i.d. random variables, independent of the interarrival times and the vacation durations. Some important performance measures such as the distribution functions and means of the virtual and the actual waiting times are given. Finally, a numerical example is presented.

scholarly journals Analysis of Queue-Length Dependent Vacations and P-Limited Service in BMAP/G/1/N Systems: Stationary Distributions and Optimal Control

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
A. D. Banik
Keyword(s):  
Queue Length ◽  
Minimum Cost ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Finite Buffer ◽  
Batch Markovian Arrival Process ◽  
Special Cases ◽  
Number Of Customers ◽  
Limited Service

We consider a finite-buffer single server queueing system with queue-length dependent vacations where arrivals occur according to a batch Markovian arrival process (BMAP). The service discipline is P-limited service, also called E-limited with limit variation (ELV) where the server serves until either the system is emptied or a randomly chosen limit of L customers has been served. Depending on the number of customers present in the system, the server will monitor his vacation times. Queue-length distributions at various epochs such as before, arrival, arbitrary and after, departure have been obtained. Several other service disciplines like Bernoulli scheduling, nonexhaustive service, and E-limited service can be treated as special cases of the P-limited service. Finally, the total expected cost function per unit time is considered to determine locally optimal values N* of N or a maximum limit L^* of L^ as the number of customers served during a service period at a minimum cost.

scholarly journals Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: D-$MAP/G_n^{(a,b)}/1$

2021 ◽  
Author(s):  
Umesh Chandra Gupta ◽  
Nitin Kumar ◽  
Sourav Pradhan ◽  
Farida Parvez Barbhuiya ◽  
Mohan L Chaudhry
Keyword(s):  
Generating Function ◽  
Discrete Time ◽  
Service Time ◽  
Batch Size ◽  
Arrival Process ◽  
General Distribution ◽  
Complete Analysis ◽  
Service Rate ◽  
Single Server ◽  
Digital Platforms

Discrete-time queueing models find a large number of applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems and computer networks. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain the vector generating function at the departure epoch. The generating function is in turn used to extract the joint distribution of queue and server content in terms of the roots of the characteristic equation. Further, we develop the relationship between the distribution at the departure epoch and the distribution at arbitrary, pre-arrival and outside observer's epochs, where the first is used to obtain the latter ones. We evaluate some essential performance measures of the system and also discuss the computing process extensively which is demonstrated by some numerical examples.

Performance analysis of the discrete-time GI/Geom/1/N queue

1996 ◽  
Vol 33 (01) ◽  
pp. 239-255 ◽  
Author(s):  
M. L. Chaudhry ◽  
U. C. Gupta
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Interarrival Time ◽  
Single Server ◽  
Finite Buffer ◽  
Supplementary Variable ◽  
Time Analysis ◽  
Variable Technique ◽  
Early Arrival ◽  
Finite Buffer Queue

This paper presents an analysis of the single-server discrete-time finite-buffer queue with general interarrival and geometric service time,GI/Geom/1/N. Using the supplementary variable technique, and considering the remaining interarrival time as a supplementary variable, two variations of this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both cases, steady-state distributions for outside observers as well as at random and prearrival epochs have been obtained. The waiting time analysis has also been carried out. Results for theGeom/G/1/Nqueue with LAS-DA have been obtained from theGI/Geom/1/Nqueue with EAS. We also give various performance measures. An algorithm for computing state probabilities is given in an appendix.

Performance analysis of the discrete-time GI/Geom/1/N queue

1996 ◽  
Vol 33 (1) ◽  
pp. 239-255 ◽  
Author(s):  
M. L. Chaudhry ◽  
U. C. Gupta
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Interarrival Time ◽  
Single Server ◽  
Finite Buffer ◽  
Supplementary Variable ◽  
Time Analysis ◽  
Variable Technique ◽  
Early Arrival ◽  
Finite Buffer Queue

This paper presents an analysis of the single-server discrete-time finite-buffer queue with general interarrival and geometric service time, GI/Geom/1/N. Using the supplementary variable technique, and considering the remaining interarrival time as a supplementary variable, two variations of this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both cases, steady-state distributions for outside observers as well as at random and prearrival epochs have been obtained. The waiting time analysis has also been carried out. Results for the Geom/G/1/N queue with LAS-DA have been obtained from the GI/Geom/1/N queue with EAS. We also give various performance measures. An algorithm for computing state probabilities is given in an appendix.

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