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 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.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (4) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

scholarly journals A discrete single server queue with Markovian arrivals and phase type group services

1995 ◽  
Vol 8 (2) ◽  
pp. 151-176 ◽  
Author(s):  
Attahiru Sule Alfa ◽  
K. Laurie Dolhun ◽  
S. Chakravarthy
Keyword(s):  
Steady State ◽  
Queueing System ◽  
State Probability ◽  
Arrival Process ◽  
Single Server ◽  
Probability Vector ◽  
Phase Type Distribution ◽  
Steady State Probability ◽  
Phase Type ◽  
Number Of Customers

We consider a single-server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady-state probability vector is shown to be of (modified) matrix-geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady-state probability vector, and some important system performance measures are developed. The steady-state waiting time distribution is derived explicitly. Some numerical examples are presented.

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 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 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.

WAITING TIME OF M/D1,∞/1 QUEUE

2007 ◽  
Vol 24 (03) ◽  
pp. 383-399
Author(s):  
WOOJIN CHANG
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Service Time ◽  
Single Server ◽  
Poisson Arrival ◽  
Mean And Variance ◽  
Simulation Results ◽  
Set Up ◽  
The Individual ◽  
Individual Service

In this paper, we consider a single-server infinite capacity queue with Poisson arrival. The server takes all waiting customers as a batch into service, and the service time of the kth customer in a batch is a + b × k, where a and b are the set-up time and the individual service time for each customer, respectively. The steady-state mean and variance of waiting time of an arbitrary customer are obtained. Simulation results confirm our analysis.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (04) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons: (i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means. (ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means. We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR. Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

Convexity results for single-server queues and for multiserver queues with constant service times

1990 ◽  
Vol 27 (02) ◽  
pp. 465-468 ◽  
Author(s):  
Arie Harel
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Sojourn Time ◽  
Interarrival Time ◽  
Service Rate ◽  
Single Server ◽  
Multiserver Queues ◽  
Service Rates ◽  
Constant Service Times ◽  
Number Of Customers

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates. These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

Convexity results for single-server queues and for multiserver queues with constant service times

1990 ◽  
Vol 27 (2) ◽  
pp. 465-468 ◽  
Author(s):  
Arie Harel
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Sojourn Time ◽  
Interarrival Time ◽  
Service Rate ◽  
Single Server ◽  
Multiserver Queues ◽  
Service Rates ◽  
Constant Service Times ◽  
Number Of Customers

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates.These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

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