Arbitrary Epochs Research Articles

Analysis of a population model with batch Markovian arrivals influenced by Markov arrival geometric catastrophes

We consider a population model in which individuals arrive as per the batch Markovian arrival process. The population is influenced by catastrophes which occur according to Markovian arrival process and it eliminates each individual of the population in a sequential order with probability p until the one individual survives or the entire population is annihilated. We first obtain the steady-state vector generating function of the population size distribution at arbitrary epoch and then the distribution is extracted in term of roots of the associated characteristic equation. Further, we obtain the population size distribution at post-catastrophe and pre-arrival epochs. Finally, a few numerical results are given.

Analysis of Markov-modulated fluid polling systems with gated discipline

<p style='text-indent:20px;'>In this paper we present two different analytical descriptions of the fluid polling model with Markov modulated load and gated discipline. The fluid arrival to the stations is modulated by a common continuous-time Markov chain (the special case when the modulating Markov chains are independent is also included). The fluid is removed at the stations during the service period by a station dependent constant rate. <p style='text-indent:20px;'>The first analytical description is based on the relationships of steady-state fluid levels at embedded server arrival and departure epochs. We derive the steady-state vector Laplace transform of the fluid levels at the stations at arbitrary epoch and its moments. The second analytical description applies the method of supplementary variables and results in differential equations, from which the joint density function of the fluid levels can be obtained. <p style='text-indent:20px;'>We also propose computational methods for both analytical descriptions and provide numerical examples to illustrate the numeric computations.

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.

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.

A Discrete-Time GIX/Geo/1 Queue with Multiple Working Vacations Under Late and Early Arrival System

This paper studies a discrete-time batch arrival GI/Geo/1 queue where the server may take multiple vacations depending on the state of the queue/system. However, during the vacation period, the server does not remain idle and serves the customers with a rate lower than the usual service rate. The vacation time and the service time during working vacations are geometrically distributed. Keeping note of the specific nature of the arrivals and departures in a discrete-time queue, we study the model under late arrival system with delayed access and early arrival system independently. We formulate the system using supplementary variable technique and apply the theory of difference equation to obtain closed-form expressions of steady-state system content distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the associated characteristic equations. We discuss the stability conditions of the system and develop few performance measures as well. Through some numerical examples, we illustrate the feasibility of our theoretical work and highlight the asymptotic behavior of the probability distributions at pre-arrival epochs. We further discuss the impact of various parameters on the performance of the system. The model considered in this paper covers a wide class of vacation and non-vacation queueing models which have been studied in the literature.

On Finite Buffer Bulk Arrival Bulk Service Queue with Queue Length and Batch Size Dependent Service

This article deals with the mathematical modeling and steady state analysis of bulk arrival bulk service queue with finite buffer space under queue length and serving batch size dependent service policy. The service time distribution follows any general distribution and allowed to change only at service initiation instant, depending on queue length and serving batch size. A single server is providing service following general bulk service rule. We employed the supplementary variable technique and the embedded Markov chain technique to compute the joint probabilities of queue length and serving batch size at various epoch, viz., arbitrary epoch, departure epoch and pre-arrival epoch, in steady state. This paper ends with a numerical optimal control analysis to demonstrate the effect of the threshold limits for general bulk service rule on the total system cost of the system.

Discrete-time queue with batch renewal input and random serving capacity rule: $$GI^X/ Geo^Y/1$$ G I X / G e o Y / 1

In this paper, we provide a complete analysis of a discrete-time infinite buffer queue in which customers arrive in batches of random size such that the inter-arrival times are arbitrarily distributed. The customers are served in batches by a single server according to the random serving capacity rule, and the service times are geometrically distributed. We model the system via the supplementary variable technique and further use the displacement operator method to solve the non-homogeneous difference equation. The analysis done using these methods results in an explicit expression for the steady-state queue-length distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the underlying characteristic equation. Our approach enables one to estimate the asymptotic distribution at a pre-arrival epoch by a unique largest root of the characteristic equation lying inside the unit circle. With the help of few numerical results, we demonstrate that the methodology developed throughout the work is computationally tractable and is suitable for light-tailed inter-arrival distributions and can also be extended to heavy-tailed inter-arrival distributions. The model considered in this paper generalizes the previous work done in the literature in many ways.

A difference equation approach for analysing a batch service queue with the batch renewal arrival process

ABSTRACTWe consider an infinite buffer single server queue wherein customers arrive according to the batch renewal arrival process and are served in batches following the random serving capacity rule. The service-batch times follow exponential distribution. This model has been studied in the past using the embedded Markov chain technique and probability generating function. In this paper we provide an alternative yet simple methodology to carry out the whole analysis which is based on the supplementary variable technique and the theory of difference equations. The procedure used here is simple in the sense that it does not require the complicated task of constructing the transition probability matrix. We obtain explicit expressions of the steady-state system-content distribution at pre-arrival and arbitrary epochs in terms of roots of the associated characteristic equation. We also present few numerical results in order to illustrate the computational procedure.

On the distribution of an infinite-buffer queueing system with versatile bulk-service rule under batch-size-dependent service policy: M/G^(a,Y)_n/1

Batch-service queues have a wide range of noteworthy applications in wireless telecommunication to deal with the multimedia type of data, manufacturing systems, group testing procedure, etc. The knowledge of both the queue and server content distributions helps the system designer to evaluate the efficiency of the queueing system in a better way. We analyse a single server infinite-buffer batch-size-dependent service queue with Poisson arrival and versatile batch-service rule. Based on supplementary variable technique, a bivariate probability generating function, the entire spectrum of new contributions, of queue content and number in a served batch at departure epoch is derived. Moreover, we perceive the complete queue and server content distribution at departure as well as arbitrary epochs. The utility of analytical results is illustrated by the inclusion of some numerical examples, which also includes the investigation of multiple zeroes.

Efficient computational analysis of non-exhaustive service vacation queues: BMAP/R/1/N(∞) under gated-limited discipline

• A finite-buffer vacation queue with batch Markovian arrival process has been analyzed. • The server is subjected to serve under gated-limited service discipline. • Proposed analysis is based on the successive substitution and the supplementary variable method. • The results have been matched with the corresponding infinite-buffer queue. • Numerical results are presented for different service- and vacation-time distributions. This paper analyzes the finite-buffer single server queue with vacation(s). It is assumed that the arrivals follow a batch Markovian arrival process ( BMAP ) and the server serves customers according to a non-exhaustive type gated-limited service discipline. It has been also considered that the service and vacation distributions possess rational Laplace-Stieltjes transformation (LST) as these types of distributions may approximate many other distributions appeared in queueing literature. Among several batch acceptance/rejection strategies, the partial batch acceptance strategy is discussed in this paper. The service limit L (1 ≤ L ≤ N ) is considered to be fixed, where N is the buffer-capacity excluding the one in service. It is assumed that in each busy period the server continues to serve until either L customers out of those that were waiting at the start of the busy period are served or the queue empties, whichever occurs first. The queue-length distribution at vacation termination/service completion epochs is determined by solving a set of linear simultaneous equations . The successive substitution method is used in the steady-state equations embedded at vacation termination/service completion epochs. The distribution of the queue-length at an arbitrary epoch has been obtained using the supplementary variable technique. The queue-length distributions at pre-arrival and post-departure epoch are also obtained. The results of the corresponding infinite-buffer queueing model have been analyzed briefly and matched with the previous model. Net profit function per unit of time is derived and an optimal service limit and buffer-capacity are obtained from a maximal expected profit. Some numerical results are presented in tabular and graphical forms.

Analysis of a Discrete-Time Geo/G/1 Queue in a Multi-Phase Service Environment with Disasters

Abstract This paper considers a discrete-time Geo/G/1 queue in a multi-phase service environment, where the system is subject to disastrous breakdowns, causing all present customers to leave the system simultaneously. At a failure epoch, the server abandons the service and the system undergoes a repair period. After the system is repaired, it jumps to operative phase i with probability qi, i = 1, 2 ⋯, n. Using the supplementary variable technique, we obtain the distribution for the stationary queue length at the arbitrary epoch, which are then used for the computation of other performance measures. In addition, we derive the expected length of a cycle time, the generating function of the sojourn time of an arbitrary customer, and the generating function of the server’s working time in a cycle. We also give the relationship between the discrete-time queueing system to its continuous-time counterpart. Finally, some examples and numerical results are presented.

Batch Renewal Arrival Process Subject to Geometric Catastrophes

We consider a stochastic model where a population grows in batches according to renewal arrival process. The population is prone to be affected by catastrophes which occur according to Poisson process. The catastrophe starts the destruction of the population sequentially, with one individual at a time, with probability p. This process comes to an end when the first individual survives or when the entire population is eliminated. Using supplementary variable and difference equation method we obtain explicit expressions of population size distribution in steady-state at pre-arrival and arbitrary epochs, in terms of roots of the associated characteristic equation. Besides, we prove that the distribution at pre-arrival epoch is asymptotically geometric. Based on our theoretical work we present few numerical results to demonstrate the efficiency of our methodology. We also investigate the impact of different parameters on the behavior of the model through some numerical examples.

On an M/G/1 queue in random environment with Min(N, V) policy

In this paper, we analyze an M∕G∕1 queue operating in multi-phase random environment with Min(N, V) vacation policy. In operative phase i, i = 1, 2, …, n, customers are served according to the discipline of First Come First Served (FCFS). When the system becomes empty, the server takes a vacation under the Min(N, V) policy, causing the system to move to vacation phase 0. At the end of a vacation, if the server finds no customer waiting, another vacation begins. Otherwise, the system jumps from the phase 0 to some operative phase i with probability qi, i = 1, 2, …, n. And whenever the number of the waiting customers in the system reaches N, the server interrupts its vacation immediately and the system jumps from the phase 0 to some operative phase i with probability qi, i = 1, 2, …, n, too. Using the method of supplementary variable, we derive the distribution for the stationary system size at arbitrary epoch. We also obtain mean system size, the results of the cycle analysis and the sojourn time distribution. In addition, some special cases and numerical examples are presented.

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.

Analysis of an M/G/1 queue with vacations and multiple phases of operation

This paper deals with an M / G / 1 queue with vacations and multiple phases of operation. If there are no customers in the system at the instant of a service completion, a vacation commences, that is, the system moves to vacation phase 0. If none is found waiting at the end of a vacation, the server goes for another vacation. Otherwise, the system jumps from phase 0 to some operative phase i with probability $$q_i$$ , $$i = 1,2, \ldots ,n.$$ In operative phase i, $$i = 1,2, \ldots ,n$$ , the server serves customers according to the discipline of FCFS (First-come, first-served). Using the method of supplementary variables, we obtain the stationary system size distribution at arbitrary epoch. The stationary sojourn time distribution of an arbitrary customer is also derived. In addition, the stochastic decomposition property is investigated. Finally, we present some numerical results.

Stationary distributions of the R[X]/R/1 cross-correlated queue

ABSTRACTWe consider an infinite buffer single server queue wherein batch interarrival and service times are correlated having a bivariate mixture of rational (R) distributions, where R denotes the class of distributions with rational Laplace–Stieltjes transform (LST), i.e., ratio of a polynomial of degree at most n to a polynomial of degree n. The LST of actual waiting time distribution has been obtained using Wiener–Hopf factorization of the characteristic equation. The virtual waiting time, idle period (actual and virtual) distributions, as well as inter-departure time distribution between two successive customers have been presented. We derive an approximate stationary queue-length distribution at different time epochs using the Markovian assumption of the service time distribution. We also derive the exact steady-state queue-length distribution at an arbitrary epoch using distributional form of Little’s law. Finally, some numerical results have been presented in the form of tables and figures.

On the GI/M/1 Queue with Vacations and Multiple Service Phases

This paper considers a GI/M/1 queue with vacations and multiple service phases. Whenever the system becomes empty, the server takes a vacation, causing the system to move to vacation phase 0. If the server returns from a vacation to find no customer waiting, another vacation begins. Otherwise, the system jumps from phase 0 to some service phase i with probability qi, i=1,2,…,N. Using the matrix geometric solution method and semi-Markov process, we obtain the distributions of the stationary system size at both arrival and arbitrary epochs. The distribution of the stationary waiting time of an arbitrary customer is also derived. In addition, we present some performance measures such as mean waiting time of an arbitrary customer, mean length of the type-i cycle, and mean number of customers in the system at the end of phase 0. Finally, some numerical examples are presented.

Queue size dependent service in bulk arrival queueing system with server loss and vacation break-off

In this paper, server provides service in two service modes, as single or in bulk according to the queue length. The server provides single service, if the queue length reaches value 'a' and batch service, if the queue length reaches the value 'c' (c > a). After completing the service if the queue length is less than 'a' then the server leaves for vacation. During secondary job (vacation) if the queue length attains the value 'a' then the server breaks the vacation and provides primary service. In the service completion epoch if the server is not reliable then the server has undergone renovation of service station. For the proposed model probability generating function of queue size at an arbitrary time epoch is obtained. Various performance measures are derived with suitable numerical illustration. A cost model is also presented for the proposed model.

Modeling and analysis of an infinite-buffer batch-arrival queue with batch-size-dependent service: [formula omitted

This paper considers a single-server infinite-buffer queue with batch-arrival, batch-service, and generally distributed batch-size-dependent service time. Based on supplementary variable technique by considering supplementary variable as remaining service time of batch in service, a bivariate probability generating function, the entire spectrum of new contributions, of queue content and number in a served batch at departure epoch is derived. We extract the joint distribution of these quantities and present them in terms of the roots of the associated characteristic equation. In addition, departure epoch probabilities are utilized to obtain joint distribution at arbitrary epoch while the later one are employed to acquire joint distribution at pre-arrival epoch. Finally, some numerical results demonstrate the proposed procedure where occurrence of multiple roots is included. Moreover, some graphical representations show that batch-size-dependent service is more potent as compared to batch-size-independent service.

Analysis of a GI/M/1 queue in a multi-phase service environment with disasters

In this paper, we study a single server GI/M/1 queue in a multi-phase service environment with disasters, where the disasters occur only when the server is busy serving customers. Whenever a disaster occurs in an operative service phase, all present customers are forced to leave the system simultaneously, the server abandons the service and an exponential repair time is set on. After the system is repaired, the server resumes his service and moves to service phase i immediately with probability q i ,i = 1,2, ... ,N . Using the matrix analytic approach and semi-Markov process, we obtain the stationary queue length distribution at both arrival and arbitrary epochs. After introducing tagged customers and the concept of a cycle, we also derive the sojourn time distribution, the duration of a cycle, and the length of the server’s working time in a service cycle. In addition, numerical examples are presented to illustrate the impact of some critical model parameters on performance measures.