Numerical analysis of a discrete-time finite capacity queue with a burst arrival

We study a discrete-time, multi-server, finite capacity queue with a burst arrival. Once the first job of a burst arrives at the queue, the successive jobs will arrive every time slot until the last job of the burst arrives. The number of jobs and the inter-arrival time of bursts are assumed to be generally distributed, and the service time is assumed to be equal to one slot. We propose an efficient numerical method to exactly obtain the job loss probability, the waiting time distribution and the mean queue length using an embedded Markov chain at the arrival instants of bursts.     

The MMCPP/GE/c Queue

We obtain the queue length probability distribution at equilibrium for a multi-server, single queue with generalised exponential (GE) service time distribution and a Markov modulated compound Poisson arrival process (MMCPP) – i.e., a Poisson point process with bulk arrivals having geometrically distributed batch size whose parameters are modulated by a Markovian arrival phase process. This arrival process has been considered appropriate in ATM networks and the GE service times provide greater flexibility than the more conventionally assumed exponential distribution. The result is exact and is derived, for both infinite and finite capacity queues, using the method of spectral expansion applied to the two dimensional (queue length by phase of the arrival process) Markov process that describes the dynamics of the system. The Laplace transform of the interdeparture time probability density function is then obtained. The analysis therefore could provide the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes, which may be both correlated and bursty.     

The PH/PH/1 queue at epochs of queue size change

The PH/PH/1 queue is considered at embedded epochs which form the union of arrival and departure instants. This provides us with a new, compact representation as a quasi-birth-and-death process, where the order of the blocks is the sum of the number of phases in the arrival and service time distributions. It is quite easy to recover, from this new embedded process, the usual distributions at epochs of arrival, or epochs of departure, or at arbitrary instants. The quasi-birth-and-death structure allows for efficient algorithmic procedures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Matrix Product-Form Solution for an LCFS-PR Single-Server Queue with Multiple Arrival Streams Governed by a Markov Chain

This paper considers a stationary single-server queue with multiple arrival streams governed by a Markov chain, where customers are served on an LCFS preemptive-resume basis. Service times of customers from each arrival stream are generally distributed and service time distributions for different arrival streams may be different. Under these assumptions, it is shown that the stationary joint distribution of queue strings representing from which arrival stream each customer in the system arrived and remaining service times of respective customers in the system has a matrix product-form solution, where matrices constituting the solution are given in terms of the infinitesimal generator of a certain Markov chain. Compared with the previous works, the result in this paper is more general in the sense that general service time distributions are allowed, and it has the advantage of computational efficiency. Note also that the result is a natural extension of the classical result for the LCFS-PR M/G/1 queue. Further, utilizing the matrix product-form solution, we derive a new expression of the vector Laplace–Stieltjes transform of the stationary distribution of unfinished work in the work-conserving single-server queue with multiple arrival streams governed by a Markov chain, which is given by the sum of matrix-geometric series.     

Analysis of the MAP/G a ,b /1/N Queue

The Markovian arrival process (MAP) is used to represent the bursty and correlated traffic arising in modern telecommunication network. In this paper, we consider a single server finite capacity queue with general bulk service rule in which arrivals are governed by MAP and service times are arbitrarily distributed. The distributions of the number of customers in the queue at arbitrary, post-departure and pre-arrival epochs have been obtained using the supplementary variable and the embedded Markov chain techniques. Computational procedure has been given when the service time distribution is of phase type.     

A markov-modulated M/M/1 queue with group arrivals

We study anM/M/1 group arrival queue in which the arrival rate, service time distributions and the size of each group arrival depend on the state of an underlying finite-state Markov chain. Using Laplace transforms and matrix analysis, we derive the results for the queue length process, its limit distribution and the departure process. In some special cases, explicit results are obtained which are analogous to known classic results.     

Analysis of GI/M/n/n queueing system with ordered entry and no waiting line

This paper deals with a multi-server, finite-capacity queuing system with recurrent input and no waiting line. The interarrival times are arbitrarily distributed whereas service times are exponentially distributed. Moreover, the servers are heterogeneous and independent of each other. Arriving customers choose the server with the lowest index number among the empty servers. When all servers are busy at a time of an arrival, that arrival must leave the system without being served. The semi-Markov process method is used to describe this model and embedded Markov chain of the process is obtained. Furthermore, the Laplace–Stieltjes transform of the distribution of interoverflow times is derived which is the main objective of the paper. Finally, it is offered a new formulation for the loss probability which provides more efficient and rapid calculation is proposed.     

A Retrial BMAP/PH/N System

A multi-server retrial queueing model with Batch Markovian Arrival Process and phase-type service time distribution is analyzed. The continuous-time multi-dimensional Markov chain describing the behavior of the system is investigated by means of reducing it to the corresponding discrete-time multi-dimensional Markov chain. The latter belongs to the class of multi-dimensional quasi-Toeplitz Markov chains in the case of a constant retrial rate and to the class of multi-dimensional asymptotically quasi-Toeplitz Markov chains in the case of an infinitely increasing retrial rate. It allows to obtain the existence conditions for the stationary distribution and to elaborate the algorithms for calculating the stationary state probabilities.     

TheM/G/1 queue with processor sharing and its relation to a feedback queue

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1–p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.     

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.     

Markov-modulatedPH/G/1 queueing systems

We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated E_K/G/1 queueing system as a special case.     

Age Process, Workload Process, Sojourn Times, and Waiting Times in a Discrete Time SM[K]/PH[K]/1/FCFS Queue

In this paper, we study a discrete time queueing system with multiple types of customers and a first-come-first-served (FCFS) service discipline. Customers arrive according to a semi-Markov arrival process and the service times of individual customers have PH-distributions. A GI/M/1 type Markov chain for a generalized age process of batches of customers is introduced. The steady state distribution of the GI/M/1 type Markov chain is found explicitly and, consequently, the steady state distributions of the age of the batch in service, the total workload in the system, waiting times, and sojourn times of different batches and different types of customers are obtained. We show that the generalized age process and a generalized total workload process have the same steady state distribution. We prove that the waiting times and sojourn times have PH-distributions and find matrix representations of those PH-distributions. When the arrival process is a Markov arrival process with marked transitions, we construct a QBD process for the age process and the total workload process. The steady state distributions of the waiting times and the sojourn times, both at the batch level and the customer level, are obtained from the steady state distribution of the QBD process. A number of numerical examples are presented to gain insight into the waiting processes of different types of customers.AMS subject classification: 60K25, 60J10This revised version was published online in June 2005 with corrected coverdate     

Heavy traffic limit for a tandem queue with identical service times

We consider a two-node tandem queueing network in which the upstream queue is M/G/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. We investigate the amount of work in the second queue at certain embedded arrival time points, namely when the upstream queue has just emptied. We focus on the case of infinite-variance service times and obtain a heavy traffic process limit for the embedded Markov chain.     

Exact Convergence Rate for the Distributions of GI/M/c/K Queue as K Tends to Infinity

We obtain the exact convergence rate of the stationary distribution ^(K) of the embedded Markov chain in GI/M/c/K queue to the stationary distribution of the embedded Markov chain in GI/M/c queue as K. Similar result for the time-stationary distributions of queue size is also included. These generalize Choi and Kim's results of the case c=1 by nontrivial ways. Our results also strengthen the Simonot's results [5].     

Steady state analysis of an M/G/1 queue with two phase service and Bernoulli vacation schedule under multiple vacation policy

This paper examines the steady state behaviour of a batch arrival queue with two phases of heterogeneous service along and Bernoulli schedule vacation under multiple vacation policy, where after two successive phases service or first vacation the server may go for further vacations until it finds a new batch of customer in the system. We carry out an extensive stationary analysis of the system, including existence of stationary regime, queue size distribution of idle period process, embedded Markov chain steady state distribution of stationary queue size, busy period distribution along with some system characteristics.     

Analysis of a Discrete-Time Finite-Capacity Single-Server Queue with Markovian-Arrival Process and Random-Bulk-Service: D-MAP/G Y /1/M

In this article, we consider a single-server, finite-capacity queue with random bulk service rule where customers arrive according to a discrete-time Markovian arrival process (D-MAP). The model is denoted by D-MAP/G ^Y /1/M where server capacity (bulk size for service) is determined by a random variable Y at the starting point of services. A simple analysis of this model is given using the embedded Markov chain technique and the concept of the mean sojourn time of the phase of underlying Markov chain of D-MAP. A complete solution to the distribution of the number of customers in the D-MAP/G ^Y /1/M queue, some computational results, and performance measures such as the average number of customers in the queue and the loss probability are presented.     

Analysis of the queue-length distribution for the discrete-time batch-service Geo/G/1/K queue

In this paper, we consider a discrete-time finite-capacity queue with Bernoulli arrivals and batch services. In this queue, the single server has a variable service capacity and serves the customers only when the number of customers in system is at least a certain threshold value. For this queue, we first obtain the queue-length distribution just after a service completion, using the embedded Markov chain technique. Then we establish a relationship between the queue-length distribution just after a service completion and that at a random epoch, using elementary ‘rate-in = rate-out’ arguments. Based on this relationship, we obtain the queue-length distribution at a random (as well as at an arrival) epoch, from which important performance measures of practical interest, such as the mean queue length, the mean waiting time, and the loss probability, are also obtained. Sample numerical examples are presented at the end.     

Quasi-birth-and-death Markov processes with a tree structure and the MMAP[K]/PH[K]/N/LCFS non-preemptive queue

This paper studies a multi-server queueing system with multiple types of customers and last-come-first-served (LCFS) non-preemptive service discipline. First, a quasi-birth-and-death (QBD) Markov process with a tree structure is defined and some classical results of QBD Markov processes are generalized. Second, the MMAP[K]/PH[K]/N/LCFS non-preemptive queue is introduced. Using results of the QBD Markov process with a tree structure, explicit formulas are derived and an efficient algorithm is developed for computing the stationary distribution of queue strings. Numerical examples are presented to show the impact of the correlation and the pattern of the arrival process on the queueing process of each type of customer.     

Analysis of customers’ impatience in queues with server vacations

Many models for customers impatience in queueing systems have been studied in the past; the source of impatience has always been taken to be either a long wait already experienced at a queue, or a long wait anticipated by a customer upon arrival. In this paper we consider systems with servers vacations where customers’ impatience is due to an absentee of servers upon arrival. Such a model, representing frequent behavior by waiting customers in service systems, has never been treated before in the literature. We present a comprehensive analysis of the single-server, M/M/1 and M/G/1 queues, as well as of the multi-server M/M/c queue, for both the multiple and the single-vacation cases, and obtain various closed-form results. In particular, we show that the proportion of customer abandonments under the single-vacation regime is smaller than that under the multiple-vacation discipline. This work was supported by the Euro-Ngi network of excellence.     

A Polynomial Factorization Approach for the Discrete Time GIX/>G/1/K Queue

This paper proposes a polynomial factorization approach for queue length distribution of discrete time GI ^X /G/1 and GI _X /G/1/K queues. They are analyzed by using a two-component state model at the arrival and departure instants of customers. The equilibrium state-transition equations of state probabilities are solved by a polynomial factorization method. Finally, the queue length distributions are then obtained as linear combinations of geometric series, whose parameters are evaluated from roots of a characteristic polynomial.