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 The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

1998 ◽  
Vol 40 (2) ◽  
pp. 207-221 ◽  
Author(s):  
Yang Woo Shin ◽  
Chareles E. M. Pearce
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Server Vacation ◽  
Batch Markovian Arrival Process ◽  
Queue Length Distribution ◽  
Special Cases ◽  
Number Of Customers

AbstractWe treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution

scholarly journals Complete analysis of MAP/G/1/N queue with single (multiple) vacation(s) under limited service discipline

2005 ◽  
Vol 2005 (3) ◽  
pp. 353-373 ◽  
Author(s):  
U. C. Gupta ◽  
A. D. Banik ◽  
S. S. Pathak
Keyword(s):  
Arrival Process ◽  
Complete Analysis ◽  
Model Parameters ◽  
Service Discipline ◽  
Single Server ◽  
Finite Buffer ◽  
Computer Communication ◽  
Multiple Vacation ◽  
Mean Waiting Time ◽  
Number Of Customers

We consider a finite-buffer single-server queue with Markovian arrival process (MAP) where the server serves a limited number of customers, and when the limit is reached it goes on vacation. Both single- and multiple-vacation policies are analyzed and the queue length distributions at various epochs, such as pre-arrival, arbitrary, departure, have been obtained. The effect of certain model parameters on some important performance measures, like probability of loss, mean queue lengths, mean waiting time, is discussed. The model can be applied in computer communication and networking, for example, performance analysis of token passing ring of LAN and SVC (switched virtual connection) of ATM.

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 an MAP/PH/1 Queue with Flexible Group Service

2017 ◽  
Vol 27 (1) ◽  
pp. 119-131 ◽  
Author(s):  
Arianna Brugno ◽  
Ciro D’Apice ◽  
Alexander Dudin ◽  
Rosanna Manzo
Keyword(s):  
Markov Chain ◽  
Service Time ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Phase Type Distribution ◽  
Stationary Probability Distribution ◽  
The Individual ◽  
Number Of Customers ◽  
Individual Service

Abstract A novel customer batch service discipline for a single server queue is introduced and analyzed. Service to customers is offered in batches of a certain size. If the number of customers in the system at the service completion moment is less than this size, the server does not start the next service until the number of customers in the system reaches this size or a random limitation of the idle time of the server expires, whichever occurs first. Customers arrive according to a Markovian arrival process. An individual customer’s service time has a phase-type distribution. The service time of a batch is defined as the maximum of the individual service times of the customers which form the batch. The dynamics of such a system are described by a multi-dimensional Markov chain. An ergodicity condition for this Markov chain is derived, a stationary probability distribution of the states is computed, and formulas for the main performance measures of the system are provided. The Laplace–Stieltjes transform of the waiting time is obtained. Results are numerically illustrated.

scholarly journals On the Queue Length Distribution in BMAP Systems

2017 ◽  
Vol 2 (4) ◽  
pp. 275 ◽  
Author(s):  
Andrzej Chydzinski
Keyword(s):  
Poisson Process ◽  
Queue Length ◽  
Length Distribution ◽  
Compound Poisson Process ◽  
Arrival Process ◽  
Finite Buffer ◽  
Batch Markovian Arrival Process ◽  
Queue Length Distribution ◽  
Markovian Processes ◽  
The Laplace Transform

Batch Markovian Arrival Process – BMAP – is a teletraffic model which combines high ability to imitate complexstatistical behaviour of network traces with relative simplicity in analysis and simulation. It is also a generalization of a wide class of Markovian processes, a class which in particular include the Poisson process, the compound Poisson process, the Markovmodulated Poisson process, the phase-type renewal process and others. In this paper we study the main queueing performance characteristic of a finite-buffer queue fed by the BMAP, namely the queue length distribution. In particular, we show a formula for the Laplace transform of the queue length distribution. The main benefit of this formula is that it may be used to obtain both transient and stationary characteristics. To demonstrate this, several numerical results are presented.

scholarly journals Optimization of the service strategy in a queueing system with energy harvesting and customers’ impatience

2016 ◽  
Vol 26 (2) ◽  
pp. 367-378 ◽  
Author(s):  
Alexander Dudin ◽  
Moon Ho Lee ◽  
Sergey Dudin
Keyword(s):  
Markov Chain ◽  
Queueing System ◽  
Threshold Value ◽  
Arrival Process ◽  
Sleep Mode ◽  
Single Server ◽  
Finite Buffer ◽  
Phase Type Distribution ◽  
New Strategy ◽  
Number Of Customers

Abstract A single-server queueing system with an infinite buffer is considered. The service of a customer is possible only in the presence of at least one unit of energy, and during the service the number of available units decreases by one. New units of energy arrive in the system at random instants of time if the finite buffer for maintenance of energy is not full. Customers are impatient and leave the system without service after a random amount of waiting time. Such a queueing system describes, e.g., the operation of a sensor node which harvests energy necessary for information transmission from the environment. Aiming to minimize the loss of customers due to their impatience (and maximize the throughput of the system), a new strategy of control by providing service is proposed. This strategy suggests that service temporarily stops if the number of customers or units of energy in the system becomes zero. The server is switched off (is in sleep mode) for some time. This time finishes (the server wakes up) if both the number of customers in the buffer and the number of energy units reach some fixed threshold values or when the number of energy units reaches some threshold value and there are customers in the buffer. Arrival flows of customers and energy units are assumed to be described by an independent Markovian arrival process. The service time has a phase-type distribution. The system behavior is described by a multi-dimensional Markov chain. The generator of this Markov chain is derived. The ergodicity condition is presented. Expressions for key performance measures are given. Numerical results illustrating the dependence of a customer’s loss probability on the thresholds defining the discipline of waking up the server are provided. The importance of the account of correlation in arrival processes is numerically illustrated.

Finite capacity vacation models with non-renewal input

1991 ◽  
Vol 28 (01) ◽  
pp. 174-197 ◽  
Author(s):  
C. Blondia
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Length Distribution ◽  
Time Instant ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Waiting Room ◽  
Special Cases

This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival epochs.

GeoX/G/1/N Queue with Vacation and Limited Service Discipline

2013 ◽  
Vol 756-759 ◽  
pp. 2470-2474
Author(s):  
Mian Zhang
Keyword(s):  
Queue Length ◽  
Batch Arrival ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queue ◽  
Arrival Times ◽  
Service Completion ◽  
Server Queue ◽  
Service Times ◽  
Limited Service

We consider a finite butter single server queue with batch arrival, where server serves a limited number of customer before going for vacation (s).The inter arrival times of batches are assumed to be independent and geometrically distribute. The service times and the vacation times of the server are generally distributed and their durations are integral multiples of slots duration. We obtain queue length distributions at service completion, vacation termination and arbitrary epochs.

Finite capacity vacation models with non-renewal input

1991 ◽  
Vol 28 (1) ◽  
pp. 174-197 ◽  
Author(s):  
C. Blondia
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Length Distribution ◽  
Time Instant ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Waiting Room ◽  
Special Cases

This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival epochs.

Functional Large Deviation Principles for Waiting and Departure Processes

1998 ◽  
Vol 12 (4) ◽  
pp. 479-507 ◽  
Author(s):  
Anatolii A. Puhalskii ◽  
Ward Whitt
Keyword(s):  
Large Deviation ◽  
Rate Function ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Departure Process ◽  
Large Deviation Principles ◽  
Special Cases ◽  
Skorohod Topologies ◽  
Departure Processes

We establish functional large deviation principles (FLDPs) for waiting and departure processes in single-server queues with unlimited waiting space and the first-in first-out service discipline. We apply the extended contraction principle to show that these processes obey FLDPs in the function space D with one of the nonuniform Skorohod topologies whenever the arrival and service processes obey FLDPs and the rate function is finite for appropriate discontinuous functions. We apply our previous FLDPs for inverse processes to obtain an FLDP for the waiting times in a queue with a superposition arrival process. We obtain FLDPs for queues within acyclic networks by showing that FLDPs are inherited by processes arising from the network operations of departure, superposition, and random splitting. For this purpose, we also obtain FLDPs for split point processes. For the special cases of deterministic arrival processes and deterministic service processes, we obtain convenient explicit expressions for the rate function of the departure process, but not more generally. In general, the rate function for the departure process evidently must be calculated numerically. We also obtain an FLDP for the departure process of completed work, which has important application to the concept of effective bandwidths for admission control and capacity planning in packet communication networks.

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