scholarly journals Analysis of a Priority Queue with Phase-Type Service and Failures

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Alexander Dudin ◽  
Sergei Dudin
Keyword(s):  
Markov Chain ◽  
Performance Measures ◽  
Priority Queue ◽  
Arrival Process ◽  
Single Server ◽  
Service Time Distribution ◽  
New Class ◽  
Phase Type ◽  
Chain Problem ◽  
Insight Into

We consider a single server queue with two types of customers. We propose a discipline of flexible priority in access that combines the features of randomization and the threshold type control. We introduce a new class of distributions, phase-type with failures (PHF) distribution, that generalizes the well-known phase-type (PH) distribution to the case when failures can occur during service of a customer. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of PHF type with the parameters depending on the type of a customer. Customers of both types can be impatient. Behavior of the system is described by the multidimensional Markov chain. Problem of existence and computation of the stationary distribution of this Markov chain is discussed in brief as well as the problem of computation of the key performance measures of the system. Numerical examples are presented that give some insight into behavior of the system performance measures under different values of the parameters defining the strategy of customers access to service.

A BMAP/PH/N SYSTEM WITH IMPATIENT REPEATED CALLS

2007 ◽  
Vol 24 (03) ◽  
pp. 293-312 ◽  
Author(s):  
VALENTINA I. KLIMENOK ◽  
DMITRY S. ORLOVSKY ◽  
ALEXANDER N. DUDIN
Keyword(s):  
Markov Chain ◽  
Time Distribution ◽  
Arrival Process ◽  
Service Time Distribution ◽  
State Distribution ◽  
Unsuccessful Attempt ◽  
Batch Markovian Arrival Process ◽  
Repeated Calls ◽  
Phase Type ◽  
Multi Server

A multi-server queueing model with a Batch Markovian Arrival Process, phase-type service time distribution and impatient repeated customers is analyzed. After any unsuccessful attempt, the repeated customer leaves the system with the fixed probability. The behavior of the system is described in terms of continuous time multi-dimensional Markov chain. Stability condition and an algorithm for calculating the stationary state distribution of this Markov chain are obtained. Main performance measures of the system are calculated. Numerical results are presented.

scholarly journals The Queueing Model MAP|PH|1|N with Feedback Operating in a Markovian Random Environment

2016 ◽  
Vol 34 (2) ◽  
Author(s):  
A.N. Dudin ◽  
A.V. Kazimirsky ◽  
V.I. Klimenok ◽  
L. Breuer ◽  
U. Krieger
Keyword(s):  
Random Environment ◽  
Queueing Systems ◽  
Arrival Process ◽  
Single Server ◽  
Service Time Distribution ◽  
Finite Buffer ◽  
Phase Type ◽  
Server Queue ◽  
Finite State ◽  
Markovian Random Environment

Queueing systems with feedback are well suited for the description of message transmission and manufacturing processes where a repeated service is required. In the present paper we investigate a rather general single server queue with a Markovian Arrival Process (MAP), Phase-type (PH) service-time distribution, a finite buffer and feedback which operates in a random environment. A finite state Markovian random environment affects the parameters of the input and service processes and the feedback probability. The stationary distribution of the queue and of the sojourn times as well as the loss probability are calculated. Moreover, Little’s law is derived.

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 Priority Queue with Many Customer Types, Correlated Arrivals and Changing Priorities

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1292
Author(s):  
Seokjun Lee ◽  
Sergei Dudin ◽  
Olga Dudina ◽  
Chesoong Kim ◽  
Valentina Klimenok
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Performance Indicators ◽  
Queueing System ◽  
Priority Queue ◽  
First Aid ◽  
Single Server ◽  
Finite Buffer ◽  
Impatient Customers ◽  
Arrival Times

A single-server queueing system with a finite buffer, several types of impatient customers, and non-preemptive priorities is analyzed. The initial priority of a customer can increase during its waiting time in the queue. The behavior of the system is described by a multi-dimensional Markov chain. The generator of this chain, having essential dependencies between the components, is derived and formulas for computation of the most important performance indicators of the system are presented. The dependence of some of these indicators on the capacity of the buffer space is illustrated. The profound effect of the phenomenon of correlation of successive inter-arrival times and variance of the service time is numerically demonstrated. Results can be used for the optimization of dispatching various types of customers in information transmission systems, emergency departments and first aid stations, perishable foods supply chains, etc.

scholarly journals Queueing Network with Moving Servers as a Model of Car Sharing Systems

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 825
Author(s):  
Chesoong Kim ◽  
Sergei Dudin ◽  
Olga Dudina
Keyword(s):  
Markov Chain ◽  
Performance Measures ◽  
Performance Indicators ◽  
Continuous Time ◽  
Queueing Network ◽  
Arrival Process ◽  
Continuous Time Markov Chain ◽  
Car Sharing ◽  
Fleet Size ◽  
Network States

We consider a queueing network with a finite number of nodes and servers moving between the nodes as a model of car sharing. The arrival process of customers to various nodes is defined by a marked Markovian arrival process. The customer that arrives at a certain node when there is no idle server (car) is lost. Otherwise, he/she is able to start the service. With known probability, which depends on the node and the number of available cars, this customer can balk the service and leave the system. The service time of a customer has an exponential distribution. Location of the server in the network after service completion is random with the known probability distribution. The behaviour of the network is described by a multi-dimensional continuous-time Markov chain. The generator of this chain is derived which allows us to compute the stationary distribution of the network states. The formulas for computing the key performance indicators of the system are given. Numerical results are presented. They characterize the dependence of some performance measures of the network and the nodes on the total number of cars (fleet size of the car sharing system) and correlation in the arrival process.

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 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 Two parallel finite queues with simultaneous services and Markovian arrivals

1997 ◽  
Vol 10 (4) ◽  
pp. 383-405 ◽  
Author(s):  
S. R. Chakravarthy ◽  
S. Thiagarajan
Keyword(s):  
Markov Process ◽  
Performance Measures ◽  
System Performance ◽  
Arrival Process ◽  
Single Server ◽  
Queueing Model ◽  
Finite Capacity ◽  
Numerical Examples ◽  
Finite Queues ◽  
Large State Space

In this paper, we consider a finite capacity single server queueing model with two buffers, A and B, of sizes K and N respectively. Messages arrive one at a time according to a Markovian arrival process. Messages that arrive at buffer A are of a different type from the messages that arrive at buffer B. Messages are processed according to the following rules: 1. When buffer A(B) has a message and buffer B(A) is empty, then one message from A(B) is processed by the server. 2. When both buffers, A and B, have messages, then two messages, one from A and one from B, are processed simultaneously by the server. The service times are assumed to be exponentially distributed with parameters that may depend on the type of service. This queueing model is studied as a Markov process with a large state space and efficient algorithmic procedures for computing various system performance measures are given. Some numerical examples are discussed.

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.

Analysis of finite-capacity polling systems

1991 ◽  
Vol 23 (2) ◽  
pp. 373-387 ◽  
Author(s):  
Hideaki Takagi
Keyword(s):  
Time Distribution ◽  
Arrival Process ◽  
Polling Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
Finite Capacity ◽  
Poisson Arrival ◽  
Finite Capacity Queues ◽  
General Service ◽  
Limited Service

We consider a system of N finite-capacity queues attended by a single server in cyclic order. For each visit by the server to a queue, the service is given continuously until that queue becomes empty (exhaustive service), given continuously only to those customers present at the visiting instant (gated service), or given to only a single customer (limited service). The server then switches to the next queue with a random switchover time, and administers the same type of service there similarly. For such a system where each queue has a Poisson arrival process, general service time distribution, and finite capacity, we find the distribution of the waiting time at each queue by utilizing the known results for a single M/G/1/K queue with multiple vacations.

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