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 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.

scholarly journals A M/M/1 Queueing Network with Classical Retrial Policy

2021 ◽  
Vol 12 (1S) ◽  
pp. 329-337
Author(s):  
S. Shanmugasundaram, Et. al.
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Queueing Network ◽  
State Probability ◽  
Queueing Model ◽  
Steady State Probability ◽  
Numerical Examples ◽  
System Length ◽  
Number Of Customers ◽  
Little's Formula

In this paper we study the M/M/1 queueing model with retrial on network. We derive the steady state probability of customers in the network, the average number of customers in the all the three nodes in the system, the queue length, system length using little’s formula. The particular case is derived (no retrial). The numerical examples are given to test the correctness of the model.

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 TheN×D-BMAP/G/1 Queueing Model: Queue Contents and Delay Analysis

2011 ◽  
Vol 2011 ◽  
pp. 1-31 ◽  
Author(s):  
Bart Steyaert ◽  
Joris Walraevens ◽  
Dieter Fiems ◽  
Herwig Bruneel
Keyword(s):  
Closed Form ◽  
Customer Service ◽  
Probability Generating Function ◽  
Mean Value ◽  
Arrival Process ◽  
Single Server ◽  
Steady State Probability ◽  
Absolute Minimum ◽  
Number Of Customers ◽  
Customer Delay

We consider a single-server discrete-time queueing system with N sources, where each source is modelled as a correlated Markovian customer arrival process, and the customer service times are generally distributed. We focus on the analysis of the number of customers in the queue, the amount of work in the queue, and the customer delay. For each of these quantities, we will derive an expression for their steady-state probability generating function, and from these results, we derive closed-form expressions for key performance measures such as their mean value, variance, and tail distribution. A lot of emphasis is put on finding closed-form expressions for these quantities that reduce all numerical calculations to an absolute minimum.

scholarly journals A single server queue under random vacation policy

2019 ◽  
Author(s):  
Priyanka kalita ◽  
Gautam Choudhury
Keyword(s):  
Steady State ◽  
Lower Cost ◽  
Queueing System ◽  
System Size ◽  
Mean Value ◽  
Single Server ◽  
Idle Period ◽  
System A ◽  
Server Queue ◽  
Number Of Customers

This paper deals with an M/G/1 queueing system with random vacation policy, in which the server takes the maximum number of random vacations till it finds minimum one message (customer) waiting in a queue at a vacation completion epoch. If no arrival occurs after completing maximum number of random vacations, the server stays dormant in the system and waits for the upcoming arrival. Here, we obtain steady state queue size distribution at an idle period completion epoch and service completion epoch. We also obtain the steady state system size probabilities and system state probabilities. Some significant measures such as a mean number of customers served during the busy period, Laplace-Stieltjes transform of unfinished work and its corresponding mean value and second moment have been obtained for the system. A cost optimal policy have been developed in terms of the average cost function to determine a locally optimal random vacation policy at a lower cost. Finally, we present various numerical results for the above system performance measures.

scholarly journals MMAP/(PH,PH)/1 Queue with Priority Loss through Feedback

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1797
Author(s):  
Divya Velayudhan Nair ◽  
Achyutha Krishnamoorthy ◽  
Agassi Melikov ◽  
Sevinj Aliyeva
Keyword(s):  
Steady State ◽  
Numerical Experiments ◽  
Queueing Systems ◽  
Priority Queue ◽  
Performance Characteristics ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Phase Type ◽  
Phase Type Distributions

In this paper, we consider two single server queueing systems to which customers of two distinct priorities (P1 and P2) arrive according to a Marked Markovian arrival process (MMAP). They are served according to two distinct phase type distributions. The probability of a P1 customer to feedback is θ on completion of his service. The feedback (P1) customers, as well as P2 customers, join the low priority queue. Low priority (P2) customers are taken for service from the head of the line whenever the P1 queue is found to be empty at the service completion epoch. We assume a finite waiting space for P1 customers and infinite waiting space for P2 customers. Two models are discussed in this paper. In model I, we assume that the service of P2 customers is according to a non-preemptive service discipline and in model II, the P2 customers service follow a preemptive policy. No feedback is permitted to customers in the P2 line. In the steady state these two models are compared through numerical experiments which reveal their respective performance characteristics.

scholarly journals A finite capacity queue with Markovian arrivals and two servers with group services

1994 ◽  
Vol 7 (2) ◽  
pp. 161-178 ◽  
Author(s):  
S. Chakravarthy ◽  
Attahiru Sule Alfa
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Queue Length ◽  
Time Distribution ◽  
Arrival Process ◽  
Finite Capacity ◽  
Waiting Time Distribution ◽  
Phase Type ◽  
Stationary Waiting Time ◽  
Number Of Customers

In this paper we consider a finite capacity queuing system in which arrivals are governed by a Markovian arrival process. The system is attended by two exponential servers, who offer services in groups of varying sizes. The service rates may depend on the number of customers in service. Using Markov theory, we study this finite capacity queuing model in detail by obtaining numerically stable expressions for (a) the steady-state queue length densities at arrivals and at arbitrary time points; (b) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. The stationary waiting time distribution is shown to be of phase type when the interarrival times are of phase type. Efficient algorithmic procedures for computing the steady-state queue length densities and other system performance measures are discussed. A conjecture on the nature of the mean waiting time is proposed. Some illustrative numerical examples are presented.

scholarly journals Retrial BMAP/PH/N Queueing System with a Threshold-Dependent Inter-Retrial Time Distribution

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 269
Author(s):  
Valentina I. Klimenok ◽  
Alexander N. Dudin ◽  
Vladimir M. Vishnevsky ◽  
Olga V. Semenova
Keyword(s):  
Exponential Distribution ◽  
Performance Indicators ◽  
Queueing System ◽  
Queueing Systems ◽  
Arrival Process ◽  
Ph Distribution ◽  
Retrial Queueing ◽  
Phase Type ◽  
Number Of Customers ◽  
Retrial Queueing Systems

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

scholarly journals Analysis of MAP/PH(1), PH(2)/2 Queue with Bernoulli Vacations

2008 ◽  
Vol 2008 ◽  
pp. 1-20 ◽  
Author(s):  
B. Krishna Kumar ◽  
R. Rukmani ◽  
V. Thangaraj
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing System ◽  
Arrival Process ◽  
Waiting Time Distribution ◽  
Steady State Probability ◽  
Matrix Geometric Method ◽  
Mean Waiting Time ◽  
The Mean ◽  
Number Of Customers

We consider a two-heterogeneous-server queueing system with Bernoulli vacation in which customers arrive according to a Markovian arrival process (MAP). Servers returning from vacation immediately take another vacation if no customer is waiting. Using matrix-geometric method, the steady-state probability of the number of customers in the system is investigated. Some important performance measures are obtained. The waiting time distribution and the mean waiting time are also discussed. Finally, some numerical illustrations are provided.

scholarly journals Analysis of an MMAP/PH1, PH2/N/∞ queueing system operating in a random environment

2014 ◽  
Vol 24 (3) ◽  
pp. 485-501 ◽  
Author(s):  
Chesoong Kim ◽  
Alexander Dudin ◽  
Sergey Dudin ◽  
Olga Dudina
Keyword(s):  
Random Environment ◽  
Queueing System ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Contact Center ◽  
Phase Type Distribution ◽  
Phase Type ◽  
Multi Server

Abstract A multi-server queueing system with two types of customers and an infinite buffer operating in a random environment as a model of a contact center is investigated. The arrival flow of customers is described by a marked Markovian arrival process. Type 1 customers have a non-preemptive priority over type 2 customers and can leave the buffer due to a lack of service. The service times of different type customers have a phase-type distribution with different parameters. To facilitate the investigation of the system we use a generalized phase-type service time distribution. The criterion of ergodicity for a multi-dimensional Markov chain describing the behavior of the system and the algorithm for computation of its steady-state distribution are outlined. Some key performance measures are calculated. The Laplace-Stieltjes transforms of the sojourn and waiting time distributions of priority and non-priority customers are derived. A numerical example illustrating the importance of taking into account the correlation in the arrival process is presented

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