Estimation of Queues at the Channel Tunnel

The Channel Tunnel, due to open in 1993, will provide a high-speed rail link between Britain and France. In this paper various scenarios are considered relating to the volume of freight which may be transported during the initial years following the opening of the tunnel. The aim is to judge whether the planned provision for freight service will be adequate in dealing with the demand. The methodology employed is that of bulk-service queueing theory; in particular, an approximate time-dependent solution to the relevant bulk-service queue is derived.     

Medicinal Inventory Control in a University Health Centre

This paper presents a stochastic, periodic-review model used to control the medicine inventories in a university health centre. Features and formulation of the model are discussed in terms of the stockout objective and the budgetary constraint. Demand and cost data used for the three medicine groups are provided. Findings of the study are analysed within the framework of sensitivity analysis, where the expected shortage levels for the medicine groups studied are taken as the performance criteria.     

Queues with postponed work

In this paper a queueing system in which work gets postponed due to finiteness of the buffer is considered. When the buffer is full (capacityK) further arrivals are directed to a pool of customers (postponed work). An arrival encountering the buffer full, will join the pool with probability γ (0<γ<1); else it is lost to the system forever. When, at a departure epoch the buffer size drops to a preassigned levelL−1 (1<L<K) or below, a postponed work is transferred with probabilityp (0<p<1) and positioned as the last among the waiting customers. If at a service completion epoch the buffer turns out to be empty and there is at least one customer in the pool, then the one ahead of all waiting in the pool gets transferred (with probability one) to the buffer and its service commences immediately. This ensures conservation of work. With arrival forming a Poisson process and service time having PH-distribution we study the long run behaviour of the system. Several system performance measures are obtained. A control problem is discussed and some numerical illustrations are provided.     

Homogeneous Customers Renege from Invisible Queues at Random Times under Deteriorating Waiting Conditions

We consider a memoryless first-come first-served queue in which customers' waiting costs are increasing and convex with time. Hence, customers may opt to renege if service has not commenced after waiting for some time. We assume a homogeneous population of customers and we look for their symmetric Nash equilibrium reneging strategy. Besides the model parameters, customers are aware only, if they are in service or not, and they recall for how long they are have been waiting. They are informed of nothing else. We show that under some assumptions on customers' utility function, Nash equilibrium prescribes reneging after random times. We give a closed form expression for the resulting distribution. In particular, its support is an interval (in which it has a density) and it has at most two atoms (at the edges of the interval). Moreover, this equilibrium is unique. Finally, we indicate a case in which Nash equilibrium prescribes a deterministic reneging time.     

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The discriminatory processor sharing queues with multiple classes of customers (abbreviated as DPS queues) are an important but difficult research direction in queueing theory, and it has many important practical applications in the fields of, such as, computer networks, manufacturing systems, transportation networks, and so forth. Recently, researchers have carried out some key work for the DPS queues. They gave the generating function of the steady-state joint queue lengths, which leads to the first two moments of the steady-state joint queue lengths. However, using the generating function to provide explicit expressions for the steady-state joint queue lengths has been a difficult and challenging problem for many years. Based on this, this paper applies the maximum entropy principle in the information theory to providing an approximate expression with high precision, and this approximate expression can have the same first three moments as those of its exact expression. On the other hand, this paper gives efficiently numerical computation by means of this approximate expression, and analyzes how the key variables of this approximate expression depend on the original parameters of this queueing system in terms of some numerical experiments. Therefore, this approximate expression has important theoretical significance to promote practical applications of the DPS queues. At the same time, not only do the methodology and results given in this paper provide a new line in the study of DPS queues, but they also provide the theoretical basis and technical support for how to apply the information theory to the study of queueing systems, queueing networks and more generally, stochastic models.     

Exponential Queues with Server Vacations

In this paper, we analyse a queueing system where the server may take a vacation. The customers arrive at the service facility according to a Poisson process, and are served if the server is available (not on vacation). We consider two models: when the server vacation cycle is independent of and dependent on the number of customers in the system. The infinitesimal generators of the underlying Markov processes have a block tri-diagonal structure, and we provide a matrix geometric solution. When the vacation cycle is independent of the customer queue length, we present a simple load-dependent approximation that is fairly accurate.     

Equivalences of Batch-Service Queues and Multi-Server Queues and Their Complete Simple Solutions in Terms of Roots

Abstract

In this article, we first present a unified discussion of several equivalence relationships among (as well as between) batch-service queues and multi-server queues, in terms of the stationary queue-length and waiting-time distributions. Then, we present a complete and simple solution for the queue-length and waiting-time distributions of the discrete-time multi-server deterministic-service Geo/D/b queue, in terms of roots of the so-called characteristic equation. This solution also represents the solutions for the other equivalent queues, as a result of the equivalence relationships. To aid in the applications of these results, sample numerical results are presented at the end.     

Martingale Methods for Analysing Single-Server Queues

In this paper we presents a martingale method for analysing queues of M/G/1 type, which have been generalised so that the system passes through a series of phases on which the service behaviour may differ. The analysis uses the process embedded at departures to create a martingale, which makes possible the calculation of the probability generating function of the stationary occupancy distribution. Specific examples are given, for instance, a model of an unreliable queueing system, and an example of a queue-length-threshold overload-control system.     

Queues with boundary assistance: the effects of truncation

We study a system of two queues with boundary assistance, represented as a continuous-time Quasi-Birth-and-Death process (QBD). Under our formulation, this QBD has a ‘doubly infinite’ number of phases. We determine the convergence norm of Neuts’ R-matrix and, consequently, the interval in which the decay rate of the infinite system can lie.     

Predicting Time-Dependent Distributions of Queues and Delays for Road Traffic at Roundabouts and Priority Junctions

Vehicle queues and delays at busy road junctions have to be treated time-dependently when the traffic demand and the available capacity are approximately equal. Existing methods allow the queue length at a given time to be directly estimated as an average over all possible evolutions of the queueing system consistent with the given initial conditions and the time-dependent arrival and service rates. The paper describes the development of methods to predict the underlying distributions. Estimates of the variance and the overall frequency distribution for queue length and delay are obtained by simulating an M/M/1 queueing model with parameters varying with time. Predictive models are developed to represent the simulation results. They require as input values of parameters describing the duration of the peak and the time-average traffic intensities and capacities.     

Routing in Queues with Delayed Information

We compare two routing-control strategies in a high-speed communication network with c parallel channels (routes), where information on service completions in down-stream servers is randomly delayed. The controller can either hold arriving messages in a common buffer, dispatching them to servers only when the delayed information becomes available (Wait option), or route jobs to the various channels, in a round-robin fashion, immediately upon their arrival. Interpreting the delays as servers's vacations and considering overall queue sizes as a measure of performance, we show that the Wait strategy is superior as long as the mean information delay is below a threshold. We calculate threshold values for various combinations of load and c and show that, for a given load, the threshold increases with c and, for fixed c, the threshold decreases with an increasing load. If information is delayed on arrival instants, rather than on service completions, we show that the system can be viewed as a tandem queue and derive a generalization of a queue-decomposition result obtained by Altman, Kofman and Yechiali.     

Joint Distributions for Interacting Fluid Queues

Motivated by recent traffic control models in ATM systems, we analyse three closely related systems of fluid queues, each consisting of two consecutive reservoirs, in which the first reservoir is fed by a two-state (on and off) Markov source. The first system is an ordinary two-node fluid tandem queue. Hence the output of the first reservoir forms the input to the second one. The second system is dual to the first one, in the sense that the second reservoir accumulates fluid when the first reservoir is empty, and releases fluid otherwise. In these models both reservoirs have infinite capacities. The third model is similar to the second one, however the second reservoir is now finite. Furthermore, a feedback mechanism is active, such that the rates at which the first reservoir fills or depletes depend on the state (empty or nonempty) of the second reservoir.The models are analysed by means of Markov processes and regenerative processes in combination with truncation, level crossing and other techniques. The extensive calculations were facilitated by the use of computer algebra. This approach leads to closed-form solutions to the steady-state joint distribution of the content of the two reservoirs in each of the models.     

Impact of Bursty Traffic on Queues

The impact of bursty traffic on queues is investigated in this paper. We consider a discrete-time single server queue with an infinite storage room, that releases customers at the constant rate of c customers/slot. The queue is fed by an M/G/∞ process. The M/G/∞ process can be seen as a process resulting from the superposition of infinitely many ‘sessions’: sessions become active according to a Poisson process; a station stays active for a random time, with probability distribution G, after which it becomes inactive. The number of customers entering the queue in the time-interval [t, t + 1) is then defined as the number of active sessions at time t (t = 0,1, ...) or, equivalently, as the number of busy servers at time t in an M/G/∞ queue, thereby explaining the terminology. The M/G/∞ process enjoys several attractive features: First, it can display various forms of dependencies, the extent of which being governed by the service time distribution G. The heavier the tail of G, the more bursty the M/G/∞ process. Second, this process arises naturally in teletraffic as the limiting case for the aggregation of on/off sources [27]. Third, it has been shown to be a good model for various types of network traffic, including telnet/ftp connections [37] and variable-bit-rate (VBR) video traffic [24]. Last but not least, it is amenable to queueing analysis due to its very strong structural properties. In this paper, we compute an asymptotic lower bound for the tail distribution of the queue length. This bound suggests that the queueing delays will dramatically increase as the burstiness of the M/G/∞ input process increases. More specifically, if the tail of G is heavy, implying a bursty input process, then the tail of the queue length will also be heavy. This result is in sharp contrast with the exponential decay rate of the tail distribution of the queue length in presence of ‘non-bursty’ traffic (e.g. Poisson-like traffic). This revised version was published online in August 2006 with corrections to the Cover Date.