scholarly journals Performance Measures of State Dependent MMPP/M/1 Queue

2018 ◽  
Vol 7 (4.10) ◽  
pp. 942 ◽  
Author(s):  
R. Sakthi ◽  
V. Vidhya ◽  
K. Mahaboob Hassain Sherieff ◽  
. .
Keyword(s):  
Performance Measures ◽  
Research Work ◽  
Arrival Process ◽  
Service Rate ◽  
Single Server ◽  
State Dependent ◽  
Markov Modulated ◽  
Service State ◽  
Modulated Process ◽  
Stationary Probability Vector

In this research work we are concerned with single unit server queue  queue with Markov Modulated process in Poisson fashion and the service time follow exponential distribution. The system is framed as a state dependent with the arrival process as Markov Modulated input and service is rendered by a single server with variation in service rate based on the intensity of service state of the system. The rate matrix that is essential to compute the stationary probability vector is obtained and various performance measures are computed using matrix method.

A Convexity Property of a Markov-Modulated Queueing Loss System

1995 ◽  
Vol 9 (2) ◽  
pp. 193-199 ◽  
Author(s):  
Charles Du ◽  
Michael Pinedo
Keyword(s):  
Transition Probability ◽  
Loss Probability ◽  
Arrival Process ◽  
Service Rate ◽  
Single Server ◽  
Convexity Property ◽  
Loss System ◽  
Markov Modulated Poisson Process ◽  
Speed Parameter ◽  
Markov Modulated

In this note we consider a single-server queueing loss system with zero buffer. The arrival process is a nonstationary Markov-modulated Poisson process. The arrival process in state i is Poisson with rate λi. The process remains in state i for a time that is exponentially distributed with rate Cαi, with c being a control or speed parameter. The service rate in state i is exponentially distributed with rate μi. The process moves from state i to state j with transition probability qij. We are interested in the loss probability as a function of c. In this note we show that, under certain conditions, the loss probability decreases when the c increases. As such, this result generalizes a result obtained earlier by Fond and Ross.

scholarly journals Computational Performance of a Single Server State Dependent Queue

2019 ◽  
Vol 8 (10) ◽  
pp. 697-700
Keyword(s):  
Performance Measures ◽  
Service Rate ◽  
Single Server ◽  
Queuing Model ◽  
Computational Performance ◽  
State Dependent ◽  
Matrix Geometric Method ◽  
Perform Sensitivity Analysis ◽  
Customer Type ◽  
Heavy Tailed

In this article we consider a single server state dependent queuing system with the service rate varying according to types of customers like normal, tagged and heavy tailed. We develop a state dependent queuing model in which the service rate depends on customer type arrive to the system with quasi-birth-death environment structure and using matrix geometric method we develop the system performance measures. Also, we use these performance measures by utilizing the maximum potential of server and perform sensitivity analysis with numerical illustrations.

scholarly journals A Taylor Series Approach for Service-Coupled Queueing Systems with Intermediate Load

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Ekaterina Evdokimova ◽  
Sabine Wittevrongel ◽  
Dieter Fiems
Keyword(s):  
Performance Measures ◽  
Queueing Systems ◽  
Poisson Processes ◽  
Series Expansions ◽  
Service Rate ◽  
Single Server ◽  
Queueing Model ◽  
State Space Explosion ◽  
Finite Queues ◽  
Service Rates

This paper investigates the performance of a queueing model with multiple finite queues and a single server. Departures from the queues are synchronised or coupled which means that a service completion leads to a departure in every queue and that service is temporarily interrupted whenever any of the queues is empty. We focus on the numerical analysis of this queueing model in a Markovian setting: the arrivals in the different queues constitute Poisson processes and the service times are exponentially distributed. Taking into account the state space explosion problem associated with multidimensional Markov processes, we calculate the terms in the series expansion in the service rate of the stationary distribution of the Markov chain as well as various performance measures when the system is (i) overloaded and (ii) under intermediate load. Our numerical results reveal that, by calculating the series expansions of performance measures around a few service rates, we get accurate estimates of various performance measures once the load is above 40% to 50%.

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.

On stability of state-dependent queues and acyclic queueing networks

1989 ◽  
Vol 21 (3) ◽  
pp. 681-701 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Single Server ◽  
General Function ◽  
State Dependent ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single server first-come-first-served queue with a stationary and ergodic input. The service rate is a general function of the workload in the queue. We provide the necessary and sufficient conditions for the stability of the system and the asymptotic convergence of the workload process to a finite stationary process at large times. Then, we consider acyclic networks of queues in which the service rate of any queue is a function of the workloads of this and of all the preceding queues. The stability problem is again studied. The results are then extended to analogous systems with periodic inputs.

Analysis of a Discrete-Time Queueing System with a Markov-Modulated input Process and constant service Rate

1996 ◽  
Vol 10 (3) ◽  
pp. 429-441 ◽  
Author(s):  
Woo-Yong Choi ◽  
Chi-Hyuck Jun
Keyword(s):  
Discrete Time ◽  
Queueing System ◽  
Service Rate ◽  
System State ◽  
Input Process ◽  
New Approach ◽  
One Dimensional ◽  
Markov Modulated ◽  
Modulated Process ◽  
Renewal Cycle

We propose a new approach to the analysis of a discrete-time queueing system whose input is generated by a Markov-modulated process and whose service rate is constant. Renewal cycles are identified and the system state on each renewal cycle is modeled as a one-dimensional Markov chain.

A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

1995 ◽  
Vol 32 (4) ◽  
pp. 1103-1111 ◽  
Author(s):  
Qing Du
Keyword(s):  
Poisson Process ◽  
Arrival Rate ◽  
Loss Probability ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Markov Modulated Poisson Process ◽  
Server Queue ◽  
Monotonicity Result ◽  
Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i. The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i. In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

A single-server queue with server vacations and a class of non-renewal arrival processes

1990 ◽  
Vol 22 (3) ◽  
pp. 676-705 ◽  
Author(s):  
David M. Lucantoni ◽  
Kathleen S. Meier-Hellstern ◽  
Marcel F. Neuts
Keyword(s):  
Waiting Times ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Arbitrary Time ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Server Queue ◽  
Arrival Processes ◽  
Markov Modulated

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes.Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

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.

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