A decomposition theorem and related results for the discriminatory processor sharing queue

In this paper, we study a discriminatory processor sharing queue with Poisson arrivals,K classes and general service times. For this queue, we prove a decomposition theorem for the conditional sojourn time of a tagged customer given the service times and class affiliations of the customers present in the system when the tagged customer arrives. We show that this conditional sojourn time can be decomposed inton+1 components if there aren customers present when the tagged customer arrives. Further, we show that thesen+1 components can be obtained as a solution of a system of non-linear integral equations. These results generalize known results about theM/G/1 egalitarian processor sharing queue.     

The customer response times in the processor sharing queue are associated

The customer response times in the egalitarian processor sharing queue are shown to be associated random variables under renewal inputs and general independent service times assumptions.The work by this author was supported in part by the National Science Foundation under grant ASC 88-8802764 and by the Office of Naval Research under grant ONR N00014-87-K-0796.     

The BMAP/PH/N retrial queue with Markovian flow of breakdowns

We consider a multi-server retrial queue with the Batch Markovian Arrival Process (BMAP). The servers are identical and independent of each other. The service time distribution of a customer by a server is of the phase (PH) type. If a group of primary calls meets idle servers the primary calls occupy the corresponding number of servers. If the number of idle servers is insufficient the rest of calls go to the orbit of unlimited size and repeat their attempts to get service after exponential amount of time independently of each other. Busy servers are subject to breakdowns and repairs. The common flow of breakdowns is the MAP. An event of this flow causes a failure of any busy server with equal probability. When a server fails the repair period starts immediately. This period has PH type distribution and does not depend on the repair time of other broken-down servers and the service time of customers occupying the working servers. A customer whose service was interrupted goes to the orbit with some probability and leaves the system with the supplementary probability. We derive the ergodicity condition and calculate the stationary distribution and the main performance characteristics of the system. Illustrative numerical examples are presented.     

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.     

A disaster queue with Markovian arrivals and impatient customers

We consider a single server queueing system in which arrivals occur according to a Markovian arrival process. The system is subject to disastrous failures at which times all customers in the system are lost. Arrivals occurring during the time the system undergoes repair are stored in a buffer of finite capacity. These customers can become impatient after waiting a random amount of time and leave the system. However, these customers do not become impatient once the system becomes operable. When the system is operable, there is no limit on the number of customers who can be admitted. The structure of this queueing model is of GI/M/1-type that has been extensively studied by Neuts and others. The model is analyzed in steady state by exploiting the special nature of this type queueing model. A number of useful performance measures along with some illustrative examples are reported.     

Calculation of output characteristics of a priority queue through a busy period analysis

In this paper, we analyze some output characteristics of a discrete-time two-class priority queue by means of probability generating functions. Therefore, we construct a Markov chain which – after analysis – provides a.o. the probability generating functions of the lengths of the busy periods of both classes. It is furthermore shown how performance measures, related to the output process, are calculated from these functions. The queueing model is kept fairly simple to explain the method of analysis of the busy periods and the output characteristics of priority queues as clearly as possible.     

Sojourn time distributions in the queue defined by a general QBD process

We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.     

The acquisition queue

We propose a new queueing model named the acquisition queue. It differs from conventional queueing models in that the server not only serves customers, but also performs acquisition of new customers. The server has to divide its energy between both activities. The number of newly acquired customers is uncertain, and the effect of the server’s acquisition efforts can only be seen after some fixed time period δ (delay). The acquisition queue constitutes a (δ+1)-dimensional Markov chain. The limiting queue length distribution is derived in terms of its probability generating function, and an exact expression for the mean queue length is given. For large values of δ the numerical procedures needed for calculating the mean queue length become computationally cumbersome. We therefore complement the exact expression with a fluid approximation. One of the key features of the acquisition queue is that the server performs more acquisition when the queue is small. Together with the delay, this causes the queue length process to show a strongly cyclic behavior. We propose and investigate several ways of planning the acquisition efforts. In particular, we propose an acquisition scheme that is designed specifically to reduce the cyclic behavior of the queue length process. This research was financially supported by the European Network of Excellence Euro-NGI. The work of the second author was supported in part by a TALENT grant from the Netherlands Organisation for Scientific Research (NWO).     

Who should be given priority in a queue?

We consider a memoryless single server queue with two classes of customers, each having its fixed entry fee. We show that profit and social welfare may benefit from a service discipline based on relative priorities.     

Analysis of the adaptive MMAP[K]/PH[K]/1 queue: A multi-type queue with adaptive arrivals and general impatience

In this paper we introduce the adaptive MMAP[K] arrival process and analyze the adaptive MMAP[K]/PH[K]/1 queue. In such a queueing system, customers of K different types with Markovian inter-arrival times and possibly correlated customer types, are fed to a single server queue that makes use of r thresholds. Service times are phase-type and depend on the type of customer in service. Type k customers are accepted with some probability a_i,k if the current workload is between threshold i − 1 and i. The manner in which the arrival process changes its state after generating a type k customer also depends on whether the customer is accepted or rejected.     

How pre-emptive priority affects completion rate in an M/M/1 queue with Poisson reneging

Queuing problems in which customers leave the queue without obtaining service (i.e. renege) have many applications. This paper looks at a queue where arrival, service and reneging events are all generated by independent Poisson processes, and customers are selected for service according to priority. A closed-form expression is derived for the probability of completing service for each priority class if high-priority customers always pre-empt low-priority ones. This result has applications in modeling the value of communications between members of an interacting population, such as a formal organization or an online community.     

The continuous spectrum for the M / M / c queue

Transient solutions for M/M/c queues are important for many purposes, in particular for staffing facilities such as call centers. In this article, we show how to use spectral analysis to find such solutions. The difficulty is that unless the number in line is bounded, one has to deal with matrices of infinite size, and hence with a countable infinite number of eigenvalues. This problem can be overcome by noting that the spectrum is dense with few exceptions. We also show how many discrete eigenvalues remain. Our theory may also work to obtain spectra for other infinite-dimensional matrices. Numerical properties of our approach are explored.     

Sojourn times in a processor sharing queue with service interruptions

We study the sojourn times of customers in an M/M/1 queue with the processor sharing service discipline and a server that is subject to breakdowns. The lengths of the breakdowns have a general distribution, whereas the on-periods are exponentially distributed. A branching process approach leads to a decomposition of the sojourn time, in which the components are independent of each other and can be investigated separately. We derive the Laplace–Stieltjes transform of the sojourn-time distribution in steady state, and show that the expected sojourn time is not proportional to the service requirement. In the heavy-traffic limit, the sojourn time conditioned on the service requirement and scaled by the traffic load is shown to be exponentially distributed. The results can be used for the performance analysis of elastic traffic in communication networks, in particular, the ABR service class in ATM networks, and best-effort services in IP networks.     

The effects of the availability of waiting-time information on a balking queue

We consider two balking queue models with different types of information about delays. Potential customers arrive according to a Poisson process, and they decide whether to stay or balk based on the available delay information. In the first model, an arriving customer learns a rough range of the current queue length. In the second model, each customer’s service time is the sum of a geometric number of i.i.d. exponential phases, and an arriving customer learns the total number of phases remaining in the system. For each information model, we compare two systems, identical except that one has more precise information. In many cases, better information increases throughput and thus benefits the service provider. But this is not always so. The effect depends on the shape of the distribution describing customers’ sensitivities to delays. We also study the effects of information on performance as seen by customers. Again, more information is often good for customers, but not always.     

Analyzing discrete-time D-BMAP/G/1/N queue with single and multiple vacations

This paper analyzes a single-server finite-buffer vacation (single and multiple) queue wherein the input process follows a discrete-time batch Markovian arrival process (D-BMAP). The service and vacation times are generally distributed and their durations are integral multiples of a slot duration. We obtain the state probabilities at service completion, vacation termination, arbitrary, and prearrival epochs. The loss probabilities of the first-, an arbitrary- and the last-customer in a batch, and other performance measures along with numerical aspects have been discussed. The analysis of actual waiting time of these customers in an accepted batch is also carried out.     

Optimal control of a queue under a quality-of-service constraint with bounded and unbounded rates

We consider a simple Markovian queue with Poisson arrivals and exponential service times for jobs. The controller chooses state-dependent service rates from an action space. The queue has a finite buffer, and when full, new jobs get rejected. The controller’s objective is to choose optimal service rates that meet a quality-of-service constraint. We solve this problem analytically and compute it numerically under two cases: When the action space is unbounded and when it is bounded.     

The asymptotic variance of a time average in a birth-death process

The variance of a time average is important for planning, running and interpreting experiments. This paper derives a simple method to find this variance for the case of a Markov process. This method is then applied to obtain the variance of a time average for the case of a birth-death process.     

Analysis of MAP/PH1,PH2/1 queue with vacations and optional secondary services

In this paper we study a queueing model in which the customers arrive according to a Markovian arrival process (MAP). There is a single server who offers services on a first-come-first-served basis. With a certain probability a customer may require an optional secondary service. The secondary service is provided by the same server either immediately (if no one is waiting to receive service in the first stage) or waits until the number waiting for such services hits a pre-determined threshold. The model is studied as a QBD-process using matrix-analytic methods and some illustrative examples are discussed.     

Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs

We consider the Markovian single-server queue that alternates between on and off periods. Upon arriving, the customers observe the queue length and decide whether to join or balk. We derive equilibrium threshold balking strategies in two cases, according to the information for the server’s state.     

Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes

We consider a Markovian queue subject to Poisson generated catastrophes. Whenever a catastrophe occurs, all customers are forced to abandon the system, the server is rendered inoperative and an exponential repair time is set on. We assume that the arriving customers decide whether to join the system or balk, based on a natural reward-cost structure. We study the balking behavior of the customers and derive the corresponding Nash equilibrium and social optimal strategies.