Actual Waiting Time Distribution Research Articles

Analyzing of a finite buffer queue with finite number of vacation policy and correlated arrivals

This paper analyses a MAP/G/1/N queue having finite number of vacations. The server takes a finite number (say J ≥ 0) of vacations whenever the system becomes empty at service completion epoch. If no clients are found by the end of the Jth vacation, the server does not go for vacation and stays in the system (called dormant period) until one client arrives. The number of vacations being finite and the server can utilise vacation periods for any other jobs. This is obvious that J = 1 and J → 1 lead to single and multiple vacation models, respectively. This research work mainly focuses more generalised vacation policy and different use cases. The following results have been obtained: 1) the distributions of clients in the queue at various epochs; 2) the Laplace-Stieltjes transform of the actual waiting-time distribution in the queue of a client under the FCFS discipline. The numerical data and graphs are presented to establish the analytical result.

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.

Multi-class M/PH/1 queues with deterministic impatience times

ABSTRACTThis article considers computational procedures for the waiting time and queue length distributions in stationary multi-class first-come, first-served single-server queues with deterministic impatience times. There are several classes of customers, which are distinguished by deterministic impatience times (i.e., maximum allowable waiting times). We assume that customers in each class arrive according to an independent Poisson process and a single server serves customers on a first-come, first-served basis. Service times of customers in each class are independent and identically distributed according to a phase-type distribution that may differ for different classes. We first consider the stationary distribution of the virtual waiting time and then derive numerically feasible formulas for the actual waiting time distribution and loss probability. We also analyze the joint queue length distribution and provide an algorithmic procedure for computing the probability mass function of the stationary joint queue length.

Waiting-time analysis of D-$${ BMAP}{/}G{/}1$$BMAP/G/1 queueing system

This paper presents a simple procedure to evaluate the actual waiting-time distributions for the first- and an arbitrary-customer of an arrival batch in the D-BMAP/G / 1 queueing system. The analysis is based on the remaining service time as the supplementary variable and the roots of the associated characteristic equation of the vector-generating function of the actual waiting-time distribution. Numerical aspects have been tested for a variety of service-time distribution and a sample of numerical outputs is presented.

MAP/M/c and M/PH/c queues with constant impatience times

This paper considers stationary MAP/M/c and M/PH/c queues with constant impatience times. In those queues, waiting customers leave the system without receiving their services if their elapsed waiting times exceed a predefined deterministic threshold. For the MAP/M/c queue with constant impatience times, Choi et al. (Math Oper Res 29:309---325, 2004) derive the virtual waiting time distribution, from which the loss probability and the actual waiting time distribution are obtained. We first refine their result for the virtual waiting time and then derive the stationary queue length distribution. We also discuss the computational procedure for performance measures of interest. Next we consider the stationary M/PH/c queue with constant impatience times and derive the loss probability, the waiting time distribution, and the queue length distribution. Some numerical results are also provided.

DISCRETE-TIME GIX/GEO/1/N QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTION

In this paper, we study a discrete-time finite buffer batch arrival queue with multiple geometric working vacations and vacation interruption: the server serves the customers at the lower rate rather than completely stopping during the vacation period and can come back to the normal working level once there are customers after a service completion during the vacation period, i.e., a vacation interruption happens. The service times during a service period, service times during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer's observation epochs. We also present probability generation function (p.g.f.) of actual waiting-time distribution in the system and some performance measures.

Analysis of discrete-time queues with batch renewal input and multiple vacations

This paper analyzes a discrete-time multiple vacations finite-buffer queueing system with batch renewal input in which inter-arrival time of batches are arbitrarily distributed. Service and vacation times are mutually independent and geometrically distributed. The server takes vacations when the system does not have any waiting jobs at a service completion epoch or a vacation completion epoch. The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system. Using the supplementary variable and the imbedded Markov chain techniques, the authors obtain the queue-length distributions at pre-arrival, arbitrary and outside observer’s observation epochs for partial-batch rejection policy. The blocking probability of the first-, an arbitraryand the last-job in a batch have been discussed. The analysis of actual waiting-time distributions measured in slots of the first-, an arbitrary- and the last-job in an accepted batch, and other performance measures along with some numerical results have also been investigated.

Analysis of a discrete-time GI/Geo/1/N queue with multiple working vacations

This paper analyzes a finite-buffer renewal input single server discrete-time queueing system with multiple working vacations. The server works at a different rate rather than completely stopping working during the multiple working vacations. The service times during a service period, service time during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. The analysis of actual waiting-time distribution and some performance measures are carried out. We present some numerical results and discuss special cases of the model.

Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection

This paper analyzes a finite buffer bulk arrival queueing system with a single workingvacation and partial batch rejection in which the inter-arrival and service times are, respectively,arbitrarily and exponentially distributed. Using the supplementary variable and the imbedded Markovchain techniques, we obtain the system length distributions at pre-arrival and arbitraryepochs. We also present Laplace-Stiltjes transform of the actual waiting time distribution in thesystem. Finally, several performance measures and a variety of numerical results in the form of tablesand graphs are discussed.

System-theoretical algorithmic solution to waiting times in semi-Markov queues

Markov renewal processes with matrix-exponential semi-Markov kernels provide a generic tool for modeling auto-correlated interarrival and service times in queueing systems. In this paper, we study the steady-state actual waiting time distribution in an infinite capacity single-server semi-Markov queue with the auto-correlation in interarrival and service times modeled by Markov renewal processes with matrix-exponential kernels. Our approach is based on the equivalence between the waiting time distribution of this semi-Markov queue and the output of a linear feedback interconnection system. The unknown parameters of the latter system need to be determined through the solution of a SDC (Spectral-Divide-and-Conquer) problem for which we propose to use the ordered Schur decomposition. This approach leads us to a completely matrix-analytical algorithm to calculate the steady-state waiting time which has a matrix-exponential distribution. Besides its unifying structure, the proposed algorithm is easy to implement and is computationally efficient and stable. We validate the effectiveness and the generality of the proposed approach through numerical examples.

Steady state analysis and computation of the GI[ x] / Mb/1/ L queue with multiple working vacations and partial batch rejection

This paper analyzes a finite-buffer bulk-arrival bulk-service queueing system with multiple working vacations and partial batch rejection in which the inter-arrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the embedded Markov chain techniques, we obtain the waiting queue-length distributions at pre-arrival and arbitrary epochs. We also present Laplace–Stiltjes transform of the actual waiting-time distribution in the queue. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.

The analysis of a finite-buffer general input queue with batch arrival and exponential multiple vacations

Vacation queueing models have wide range of application in several areas including computer-communication, and manufacturing systems. A finite-buffer single-server queue with renewal input and multiple exponential vacations has been analysed by Karaesmen and Gupta (1996). In this paper we extend the analysis to cover the batch arrivals, i.e. we consider a batch arrival single-server queue with renewal input and multiple exponential vacations. Using the imbedded Markov chain and supplementary variable techniques we obtain steady-state distribution of number of customers in the system at pre-arrival and arbitrary epochs. The Laplace-Stieltjes transforms of the actual waiting-time distribution of the first-, arbitrary- and last-customer of a batch under First-Come-First-Serve discipline have been derived. Finally, we present useful performance measures of interest such as probability of blocking, average queue (system) length. Some tables and graphs showing the effect of model parameters on key performance measures are presented.

Analyzing a multiserver bulk-service finite-buffer queue

This paper analyzes a finite-buffer multiserver bulk-service queueing system in which the interarrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the queue-length distributions at prearrival and arbitrary epochs. We also present Laplace–Stiltjes transform of the actual waiting-time distribution in the queue. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.

Performance analysis of a [formula omitted] queue with partial- and total-batch rejection

This paper analyzes a discrete-time multiserver finite-buffer queueing system with batch renewal arrivals in which inter-batch time of batches and service times of customers are, respectively, arbitrarily and geometrically distributed. Each customer gets service from only one server. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the state probabilities at prearrival, arbitrary and outside observer’s observation epochs for partial- and total-batch rejection policies. The blocking probability of the first-, an arbitrary- and the last-customer in a batch, and other performance measures along with some numerical results have been discussed. The analysis of actual waiting-time distributions measured in slots of the first-, an arbitrary- and the last-customer in an accepted batch has also been investigated. Finally, it is shown that in the limiting case the results obtained in this paper tend to the continuous-time counterpart.

Single-Server Queues with Markov-Modulated Arrivals and Service Speed

This paper considers single-server queues with several customer classes. Arrivals of customers are governed by the underlying continuous-time Markov chain with finite states. The distribution of the amount of work brought into the system on arrival is assumed to be general, which may differ with different classes. Further, the service speed depends on the state of the underlying Markov chain. We first show that given such a queue, we can construct the corresponding new queue with constant service speed by means of a change of time scale, and the time-average quantities of interest in the original queue are given in terms of those in the new queue. Next we characterize the joint distribution of the length of a busy period and the number of customers served during the busy period in the original queue. Finally, assuming the FIFO service discipline, we derive the Laplace--Stieltjes transform of the actual waiting time distribution in the original queue.

Analysis and Computation of the Joint Queue Length Distribution in a FIFO Single-Server Queue with Multiple Batch Markovian Arrival Streams

This paper considers a work-conserving FIFO single-server queue with multiple batch Markovian arrival streams governed by a continuous-time finite-state Markov chain. A particular feature of this queue is that service time distributions of customers may be different for different arrival streams. After briefly discussing the actual waiting time distributions of customers from respective arrival streams, we derive a formula for the vector generating function of the time-average joint queue length distribution in terms of the virtual waiting time distribution. Further assuming the discrete phase-type batch size distributions, we develop a numerically feasible procedure to compute the joint queue length distribution. Some numerical examples are provided also.

The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

A finite-capacity queue with exhaustive vacation/close-down/setup times and Markovian arrival processes

We consider a finite-capacity single-server vacation model with close-down/setup times and Markovian arrival processes (MAP). The queueing model has potential applications in classical IP over ATM or IP switching systems, where the close-down time corresponds to an inactive timer and the setup time to the time delay to set up a switched virtual connection (SVC) by the signaling protocol. The vacation time may be considered as the time period required to release an SVC or as the time during which the server goes to set up other SVCs. By using the supplementary variable technique, we obtain the queue length distribution at an arbitrary instant, the loss probability, the setup rate, as well as the Laplace–Stieltjes transforms of both the virtual and actual waiting time distributions.

Analysis of a finite buffer with non-preemptive priority scheduling

A single-server queue with finite capacity, two distinct service classes and a non-preemptive priority scheduling discipline is analyzed. The arrival streams of the different classes are independent Poisson processes. The service times follow exponential distributions depending on the service classes. To calculate the steady-state distribution of the number of customers of each class in the system, we develop a new recursive approach. Applying matrix-analytic methods, we state the waiting-time distributions of both classes and calculate the Laplace-Stieltjes transform as well as the moments of the actual waiting-time distribution of each class