System Length Distributions Research Articles

A note on [formula omitted] queueing system

This article investigates the discrete-time GeoX/G/1 queue with an early arrival system. The system length distributions at outside observer’s, post-departure and random epochs as well as the waiting time distribution of an arbitrary customer in a batch are derived. The system of difference equations developed using the supplementary variable technique allows us to directly compute the probability generating functions for these distributions. Some numerical results are provided in tabular form.

Customers' joining behavior in an unobservable <i>GI/Geo/m</i> queue

<p style='text-indent:20px;'>This paper studies the equilibrium balking strategies of impatient customers in a discrete-time multi-server renewal input queue with identical servers. Arriving customers are unaware of the number of customers in the queue before making a decision whether to join or balk the queue. We model the decision-making process as a non-cooperative symmetric game and derive the Nash equilibrium mixed strategy and optimal social strategies. The stationary system-length distributions at different observation epochs under the equilibrium structure are obtained using the roots method. Finally, some numerical examples are presented to show the effect of the information level together with system parameters on the equilibrium and social behavior of impatient customers.</p>

Synchronized abandonment in discrete-time renewal input queues with vacations

<p style='text-indent:20px;'>We consider a discrete-time infinite buffer renewal input queue with multiple vacations and synchronized abandonment. Waiting customers get impatient during the server's vacation and decide whether to take service or abandon simultaneously at the vacation completion instants. Using the supplementary variable technique and difference operator method, we obtain an explicit expression to find the steady-state system-length distributions at pre-arrival, random, and outside observer's observation epochs. We provide the stochastic decomposition structure for the number of customers and discuss the various performance measures. With the help of numerical experiments, we show that the method formulated in this work is analytically elegant and computationally tractable. The results are appropriate for light-tailed inter-arrival distributions and can also be leveraged to find heavy-tailed inter-arrival distributions.</p>

Performance analysis of renewal input queues with multiple vacations and synchronized abandonment

ABSTRACT This paper studies a single server renewal input queueing model with multiple vacations in the presence of impatient customers. The server follows an exhaustive vacation policy whenever the system is empty. During vacation, waiting customers become impatient and decide independently whether to abandon the system simultaneously or avail service at the vacation completion instant. Using the supplementary variable and difference equation techniques, we obtain the steady-state system-length distributions at pre-arrival and arbitrary epochs. We present the stochastic decomposition structure for the number of customers in the system and derive several performance measures including the sojourn time. Moreover, we examine the impact of various parameters on the system performance measures. A potential application of the model in the inventory management of an e-commerce company with fresh agricultural products is illustrated. Numerical results discuss the effects of the system parameters and provide insight into the performance of such systems.

Alternative approach based on roots for computing the stationary queue-length distributions in GIX/M(1,b)/1 single working vacation queue

The purpose of this paper is to present an alternative algorithm for computing the stationary queue-length and system-length distributions of a single working vacation queue with renewal input batch arrival and exponential holding times. Here we assume that a group of customers arrives into the system, and they are served in batches not exceeding a specific number b. Because of batch arrival, the transition probability matrix of the corresponding embedded Markov chain for the working vacation queue has no skip-free-to-the-right property. Without considering whether the transition probability matrix has a special block structure, through the calculation of roots of the associated characteristic equation of the generating function of queue-length distribution immediately before batch arrival, we suggest a procedure to obtain the steady-state distributions of the number of customers in the queue at different epochs. Furthermore, we present the analytic results for the sojourn time of an arbitrary customer in a batch by utilizing the queue-length distribution at the pre-arrival epoch. Finally, various examples are provided to show the applicability of the numerical algorithm.

Modelling and analysis of D-BMAP/D-MSP/1 queue using RG-factorization

ABSTRACT The current investigation of our paper is concentrated on an infinite-buffer single-server queueing model with discrete-time batch Markovian arrival process and Markovian service process. The steady-state behaviour of this discrete-time queueing system at various significant time epochs are presented. We first analyze the system-length distribution at outside observer’s epoch using the -factorization technique combined with the spectral expansion method to compute the stochastic matrix . For the purpose of comparative study, a comprehensive analysis of the system-length distribution at outside observer’s epoch is also carried out using the roots method. Next, we obtain the explicit closed-form expressions of the system-length distributions at pre-arrival, intermediate, post-departure and random epochs by developing the relations among them in equilibrium state. The analysis of waiting-time distribution in the queue measured in slots of an arbitrary customer in an arriving batch is also discussed. Computational experiences with adequate numerical results are conferred in the structure of tables and graphs. Finally, we suggest that our queueing model seems to be fitted for outpatient service of any specialized hospital in a smart city.

The queue revisited

We present analytic expressions (in terms of roots of the underlying characteristic equation) for the steady-state distributions of the number of customers for the finite-state queueing model with partial-batch rejection policy. We obtain the system-length distributions at a service-completion epoch by applying the imbedded Markov chain technique. Using the roots of the related characteristic equation, the method leads to giving a unified approach for solving both finite- and infinite-buffer systems. We find relationships between system-length distributions at departure, random, and arrival epochs using discrete renewal theory and conditioning on the system states. Based on these relationships, we obtain various performance measures and provide numerical results for the same. We also perform computational analysis and compare our results with respect to the solution obtained by solving a linear system of equations in terms of running times.

Stationary Analysis of an Infinite-Buffer D-MAP/D-MSP/1 Queueing System

This paper deals with the analysis of an infinite-buffer single-server D-MAP/D-MSP/1 queueing system. This queueing system can be analyzed by representing level-independent quasi-birth-and-death process in tridiagonal structure. The proposed analysis is based on the use of matrix-geometric method in conjunction with the spectral method to obtain the system-length distribution at outside observer’s epoch. We derive the stationary system-length distributions at random, prearrival, intermediate and post-departure epochs using the system-length distribution at outside observer’s epoch. The waiting-time distribution in the queue measured in slots of an arriving customer is also carried out. Computational procedures along with numerical results are provided to confirm the correctness of our analytical results.

Analysis of $GI^{[X]}/D$-$MSP/1/\infty$ queue using $RG$-factorization

<p style='text-indent:20px;'>This paper analyzes an infinite-buffer single-server queueing system wherein customers arrive in batches of random size according to a discrete-time renewal process. The customers are served one at a time under discrete-time Markovian service process. Based on the censoring technique, the UL-type <inline-formula><tex-math id="M1">\begin{document}$ RG $\end{document}</tex-math></inline-formula>-factorization for the Toeplitz type block-structured Markov chain is used to obtain the prearrival epoch probabilities. The random epoch probabilities are obtained with the help of classical principle based on Markov renewal theory. The system-length distributions at outside observer's, intermediate and post-departure epochs are obtained by making relations among various time epochs. The analysis of waiting-time distribution measured in slots of an arbitrary customer in an arrival batch has also been investigated. In order to unify the results of both discrete-time and its continuous-time counterpart, we give a brief demonstration to get the continuous-time results from those of the discrete-time ones. A variety of numerical results are provided to illustrate the effect of model parameters on the performance measures.

Modelling and analysis of a discrete-time GI^X/Geom/1/K queue with N threshold policy

This paper presents modelling and analysis of a discrete-time GIX/Geom/1/K queueing system (where K is the capacity of the system) with N threshold policy for the early arrival system (EAS). The server is turned off when the system is vacant and checks the queue length every time for an arrival of a batch. As soon as the queue length reaches a pre-specified value N(1 ≤ N ≤ K), the server turns on and serves continuously until the system becomes vacant. We obtain the steady state system length distributions at pre-arrival, arbitrary and outside observer's epochs using the combination of the supplementary and the imbedded markov chain techniques. Various performance characteristics like average number of users in the queue/system, blocking probabilities of users (first-, an arbitrary- and last-user of an arriving batch) and average waiting time are obtained analytically with numerical analysis. The numerical analysis data are presented in graphical format for blocking probabilities under different buffer size values.

Performance analysis of the GI/D-MSP/1 queue with N-policy and its optimal control

ABSTRACT This paper investigates a discrete-time GI/D-MSP/1/\\infty queueing system under N-policy with renewal input. The service process is correlated and its structure is constructed through discrete-time Markovian service process. The idle server resumes to serve the customers as soon as the number of waiting customers in the queue reaches a predefined threshold value N and serves the customers exhaustively until the system becomes empty. We use the matrix-geometric method to derive the system-length distribution at prearrival epoch. Employing the Markov renewal theory, we obtain the system-length distribution at random epoch. We also carried out the system-length distributions at outside observer’s, intermediate and post-departure epochs. Further, we find the waiting-time distribution in the queue measured in slots of an arrival customer. An expected linear cost function per unit time is considered to determine the optimal value of N, which minimises the expected cost function. Some numerical results are demonstrated to measure the effects of N, interarrival-time distributions and other model parameters.

Analysis of renewal input batch service queue with impatient customers and multiple working vacations

ABSTRACTThis paper presents a renewal input single server batch service queue with impatient customers and multiple working vacations. An arriving customer may join the system with probability or balk with probability . The service is carried out in batches with a maximum threshold limit . The server takes a working vacation when there are no customers in the system, and continues taking multiple working vacations as long as there are no customers at working vacation completion epochs. Applying the embedded Markov chain approach and the displacement operator method, we find the probability distribution of the queue-length at pre-arrival epoch. With the help of the Markov renewal theory and semi-Markov processes, we compute the probability distribution of the queue-length at an arbitrary epoch. We also obtain the system-length distributions at both pre-arrival and arbitrary epochs and present a potential application of the model in cloud systems. Some corresponding results underneath special cases are found by putting suitable values. We present various performance measures and waiting-time distributions in the queue. Numerical results are demonstrated to emphasize the qualitative features of the model.

Performance analysis of a GI/D- $$MSP/1/\infty$$ M S P / 1 / ∞ queueing system under different service phase initiations

The main objective of this paper is to derive some analytic results concerning GI/D- $$MSP/1/\infty$$ queueing model in which customers arrive according to a discrete-time renewal process. We assume that the service process is correlated and its configuration is designed through discrete-time Markovian service process under different service phase initiations to start a busy period. The system-length distributions at various time epochs have been obtained by employing the matrix-geometric approach. An explicit closed form expression for the waiting-time distribution in the queue measured in slots has also been derived. Computational experiences with a variety of numerical results are discussed to show the effect of system parameters on the performance measures. This queueing model has potential applications in the domain of computer networks, telecommunication systems, and wireless local communication networks for evaluating the various system performances. An application of this type of queueing model in the domain of network coding in wireless communications is also discussed. This provides relevant information to the system analysts, researchers, and industry people who are interested to investigate congestion problems using queueing theory.

Stationary Analysis and Optimal Control Under Multiple Working Vacation Policy in a GI/M(a,b)/1 Queue

This paper considers an infinite buffer renewal input queue with multiple working vacation policy wherein customers are served by a single server according to general bulk service (a, b)-rule (1 ≤ a ≤ b). If the number of waiting customers in the system at a service completion epoch (during a normal busy period) is lower than ‘a’, then the server starts a vacation. During a vacation if the number of waiting customers reaches the minimum threshold size ‘a’, then the server starts serving this batch with a lower rate than that of the normal busy period. After completion of a batch service during working vacation, if the server finds less than ‘a’ customers accumulated in the system, then the server takes another vacation, otherwise the server continues to serve the available batch with that lower rate. The maximum allowed size of a batch in service is ‘b’. The authors derive both queue-length and system-length distributions at pre-arrival epoch using both embedded Markov chain approach and the roots method. The arbitrary epoch probabilities are obtained using the classical argument based on renewal theory. Several performance measures like average queue and system-length, mean waiting-time, cost and profit optimization are studied and numerically computed.

Ant colony optimisation for impatient customer queue under N-policy and Bernoulli schedule vacation interruption

This paper analyses a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption under N-policy. Arriving customer decide either to join the system or to balk. After joining the queue, the customer may renege. The server leaves for working vacation after serving the queue exhaustively. At a service completion instant during working vacation, the server continue the vacation with probability q or if there are at least N customers in the queue the server interrupts the vacation and resumes regular service with probability 1 − q. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed and are mutually independent. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimisation are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

Ant colony optimisation for impatient customer queue under N-policy and Bernoulli schedule vacation interruption

This paper analyses a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption under N-policy. Arriving customer decide either to join the system or to balk. After joining the queue, the customer may renege. The server leaves for working vacation after serving the queue exhaustively. At a service completion instant during working vacation, the server continue the vacation with probability q or if there are at least N customers in the queue the server interrupts the vacation and resumes regular service with probability 1 &minus; q. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed and are mutually independent. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimisation are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

Stationary analysis of a BMAP/R/1 queue with R-type multiple working vacations

ABSTRACTWe consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple working vacation policy. The service time during a working vacation is generally distributed random variable which is independent of the service times during a normal busy period as well as the arrival process. Duration of service times during a normal busy period and duration of working vacation times follow the class of distributions whose Laplace-Stieltjes transforms are rational functions (R-type distributions). The service time during a normal busy period, working vacation time, and the service time during a working vacation are independent of each other as well as of the arrival process. If a working vacation terminates while service is going on for a customer at head of the queue in vacation mode then, the server switches to normal mode and the customer at head of the queue is entitled to receive a full service time in the normal busy period irrespective of the amount of service received by the customer at head of the queue during the previous working vacation period. We obtain system-length distributions at various epoch, such as post-departure, pre-arrival, arbitrary, and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain system-length distribution at post-departure epoch. Later, we use supplementary variable technique and simple algebraic manipulations to obtain system-length distribution at arbitrary epoch using the system-length distribution at post-departure epoch. Some important performance measures, such as mean system lengths and mean waiting time have been obtained. Finally, some numerical results have been presented in the form of tables and graphs to show the applicability of the results obtained in this article. The model has potential application in areas of computer and communication networks, such as ethernet passive optical network (EPON).

A simple analysis of system characteristics in the batch service queue with infinite-buffer and Markovian service process using the roots method:GI/C-MSP(a,b)/1/∞

We consider an infinite-buffer single-server queue with renewal input and Markovian service process where customers are served in batches according to a general bulk service rule. Queue-length distributions at epochs of pre-arrival, arbitrary and post-departure have been obtained along with some important performance measures such as mean queue lengths and mean waiting times in both the system as well as the queue. We also obtain the steady-state service batch size distributions as well as system-length distributions. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of queue-length distribution at a pre-arrival epoch. Also, we provide analytical and numerical comparison between the roots method used in this paper and the matrix geometric method in terms of computational complexities and required computation time to evaluate pre-arrival epoch probabilities for both the methods. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.

Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging

This paper analyzes a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption. Arriving customers decide either to join the system or to balk. After joining the queue the customers may renege based on their desire for service or their unwillingness for waiting. At a service completion instant during working vacations, the server can decide either to continue the vacation with probability $q$ or interrupt the vacation and resume regular service period with probability $1-q$. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimization are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

An alternative method for computing system-length distributions of BMAP/R/1 and BMAP/D/1 queues using roots

In this paper, we present closed-form expressions for system-length distributions in terms of roots outside the unit disk of the characteristic equation of the BMAP/R/1 queue, where arrival process is batch Markovian arrival process (BMAP) and R represents a class of distributions having rational Laplace–Stieltjes transform. The unknown boundary vector has been evaluated using the roots (inside and on the unit disk) of the characteristic equation. Several numerical results are presented for a variety of arrival and service-time distributions including phase-type (PH) and matrix-exponential (ME) which cover a wide variety of distributions that arise in applications. Using an approximated representation of ME distribution, results for BMAP/D/1 queue are also presented. We compared our result with the results obtained using classical matrix analytic method as well as cyclic reduction algorithm. It is shown that the computation-time of the proposed method does not depend upon the parameters like traffic intensity and correlation co-efficient. The method is analytically quite simple and easy to implement.