Expected Queue Length Research Articles

Minimal waiting times in static traffic control

The paper discusses the question of the optimal control of an unsymmetric bottleneck system with Poisson arrival processes having the minimization of the mean individual waiting time as objective. The setup allows the straightforward generalization to more complicated forms of traffic organization. The notion of the mean individual waiting time is based on a theorem of the Little type, which is derived by a strong law of large numbers. The proof makes use of McNeil's formula, which connects the expected total waiting time with the expected queue length.

Queue Length Analysis of Non-Preemptive DiffServ Networks

Abstract Considering that differential services (DiffServ) networking technologies are under development to support diverse quality-of-service (QoS) requirements, this letter provides the upper and lower bounds of the expected queue length for each priority class in a non-preemptive priority queueing system. The results illustrate the possible performance gain in terms of buffer requirement and/or waiting time that Diffserv networks can provide compared to those that do not support service differentiation.

Stochastic online binpacking problem

We consider the following stochastic bin packing process: the items of different sizes arrive at times t = 0, 1, 2, … and are packed into unit size bins using "largest first" rule. The unpacked items form queues. Coffman and Stolyar [3] introduced this system and posed the following question: under which conditions expected queue lengths are bounded (system is stable)? We provide exact computable conditions for stability of this system using Lyapunov function technique. The result holds for a very general class of distributions of the arrival processes.

A queueing system with single arrival bulk service and single departure

In this note, we study a Markovian queueing system with accessible batches for service, but units depart individually. This system generalizes the M/ M/ C, bulk service, and accessible batch queueing systems. We compute the system size probabilities in transient and steady states, waiting time distribution, busy period distribution, expected queue length, etc. Little's formula is shown to hold for the present set up. We investigate the optimal values of the maximum batch size for service and capacity of the system.

Queue length and occupancy in discrete-time cyclic networks with several types of nodes

For a discrete-time, closed, cyclic queueing network, where the nodes have independent, geometric service times, the equilibrium rate of local progress is determined. Faster nodes are shown to have a capacity depending only on the service probabilities. A family of such networks, each with the same number of types of nodes, is analyzed. If the number of nodes approaches infinity, and if the ratio of jobs to nodes has a positive limit and each node type has an asymptotic density, then for a given node type, the limits of the proportion of occupied nodes and the expected queue length are calculated. These values depend on the service parameter and on the asymptotic rate of local progress. The faster nodes can attain their capacity only when the limiting density of nodes of slowest type is zero.

Markovian bulk service queue with delayed vacations

Scope and purpose This model deals with the study of a general bulk service queueing system with repeated server vacation and changeover time. Whenever the server finds a−1 customers waiting in the queue, instead of going for vacation immediately, the server will wait in the system for some time which is called the changeover time. If there is an arrival during the changeover time the server starts service, otherwise the server will go for a vacation. Such bulk service queueing models are useful in many real life transportation systems such as shuttle bus service, taxi stand, express elevators, tour guides and so on. An M/ M( a, b)/1 queueing system with multiple vacations is studied, in which if the number of customers in the queue is a−1 either at a service completion epoch or at a vacation completion point, the server will wait for an exponential time in the system which is called the changeover time. During this changeover time if there is an arrival the server will start service immediately, otherwise at the end of the changeover time the server will go for a vacation. The duration of vacation is also exponential. This paper is concerned with the determination of the stationary distribution of the number of customers in the queue and the waiting time distribution of an arriving customer. The expected queue length is also obtained. Sample numerical illustrations are given.

Transient solution of a queueing system with heterogeneous service rates and finite additional spaces using matrix technique

A Markovian queue with heterogeneous service rates and two types of finite spaces namely ordinary and additional space, depending upon queue length is discussed. Whenever the system size reaches a certain length, the system adds an additional finite space probabilistically. The object of this paper is to find the transient probabilities of the number of customers in the system and the expected queue length of the system. Moreover, the characteristics like steady-state probabilities of the number of customers in the system when the additional space is not available are also obtained. Associating the cost with the rent of the additional space and the profit with each customer being served, the optimum size of the ordinary space as well as the optimum number of unit size additional space to be hired and the like are obtained. The above problem is discussed with a numerical example and tabulated

A POLLING MODEL WITH RETRIAL CUSTOMERS

A polling model with n stations and switchover times is considered. The customers are of n different types, arrive to the system according to the Poisson distribution in batches of random size, and if they find the server unavailable, they start to make retrials until succeed to find a position for service. Each batch may contain customers of different types while the numbers of customers belonging to each type in a batch are distributed according to a multivariate general distribution. The server, upon polling a station, stays there for an exponential period of time and if a customer asks for service before this time expires, the customer is served and a new "stay period" begins. Finally the service times and the switchover times are both arbitrarily distributed with different distributions for the different stations. For such a model we obtain formulas for the expected number of retrial customers in each station in a steady state. Our results can be easily adapted to hold for zero switchover times and also in the case of the ordinary exhaustive service polling model with (without) switchover times and correlated batch arrivals. In all cases mentioned above (retrial model, exhaustive model, switchover times, zero switchover times) to find the expected queue lengths we need finally to solve a set of only n linear equations (O(n^3) arithmetic operations to compute the coefficients). Tables of numerical values are finally obtained and used to observe the system performance when we vary the values of the parameters.

Optimal control of arrivals to queues with delayed queue length information

Considers discrete-time versions of two classical problems in the optimal control of admission to a queueing system: (i) optimal routing of arrivals to two parallel queues and (ii) optimal acceptance/rejection of arrivals to a single queue. The authors extend the formulation of these problems to permit a k step delay in the observation of the queue lengths by the controller. For geometric inter-arrival times and geometric service times the problems are formulated as controlled Markov chains with expected total discounted cost as the minimization objective. For problem (i) the authors show that when k=1, the optimal policy is to allocate an arrival to the queue with the smaller expected queue length (JSEQ: Join the Shortest Expected Queue). The authors also show that for this problem, for k/spl ges/2, JSEQ is not optimal. For problem (ii) the authors show that when k=1, the optimal policy is a threshold policy. There are, however, two thresholds m/sub 0//spl ges/m/sub 1/>0, such that m/sub 0/ is used when the previous action was to reject, and m/sub 1/ is used when the previous action was to accept.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Asymptotic analysis for closed multiclass queueing networks in critical usage

We consider a class of closed multiclass queueing networks containing First-Come-First-Serve (FCFS) and Infinite Server (IS) stations. These networks have a productform solution for their equilibrium probabilities. We study these networks in an asymptotic regime for which the number of customers and the service rates at the FCFS stations go to infinity with the same order. We assume that the regime is in critical usage, whereby the utilizations of the FCFS servers slowly approach one. The asymptotic distribution of the normalized queue lengths is shown to be in many cases a truncated multivariate normal distribution. Traffic conditions for which the normalized queue lengths arealmost asymptotically independent are determined. Asymptotic expansions of utilizations and expected queue lengths are presented. We show through an example how to obtain asymptotic expansions of performance measures when the networks are in mixed usage and how to apply the results to networks with finite data.

A heuristic for the transient expected queue length of Markovian queueing systems

We present a heuristic for the transient expected queue length for stationary, single-server, Markovian queueing systems that begin at rest. The heuristic is closed-form, suitable for applications work, and requires knowledge only of basic system parameters and the equilibrium expected queue length. Our empirical results demonstrate the heuristic performs well over a specified portion of the transient period for systems under moderate to heavy traffic.

Perturbation analysis of the M/M/1 queue in a markovian environment via the matrix-geometric method

In this paper, we consider a family of M(t)/M/1 queues in which customers arrive according to nonhomogenous Poisson processes with intensity We assume that λt(e) is an irreducible finite-state Markov process. Based on the matrix-geometric method, we use perturbation analysis to obtain the second order approximations for the expected queue length for two cases where e is small and where e is large. Using these approximations, we show that the expected waiting times are strictly decreasing in e when e is small. In the case where e is large, we show that the expected waiting times are strictly decreasing in e if the intensity process is dynamically reversible. These results partially answer a question posed by Rolski

Performance analysis of clock-regulated queues with output multiplexing in three different 2 × 2 crossbar switch architectures

Switches in interconnection networks for highly parallel shared memory computer systems may be implemented with different internal buffer structures. For a 2 × 2 synchronous switch, previous studies have often assumed a switch composed of two queues, one at each output, each of which has unbounded size and may accept two inputs every clock cycle. Hardware implementations may actually use simpler queue designs and will have bounded size. Two additional models for a 2 × 2 switch using queues that may accept only one input at a time are analyzed. The first uses four queues, one for each input/output pair. The second uses two queues, one at each input. In each case, a multiplexer blocks one queue if two queues desire the same output, making these models more difficult to analyze than the previous model. Maximum bandwidth, expected queue length, expected waiting time, and queue length distribution are presented for both models, with unbounded queue size and with queue size equal to 1.

A note on the convexity of the expected queue length of the M/M/s queue with respect to the arrival rate: a third proof

The convexity of the expected number in an M/M/s queue with respect to the arrival rate (or traffic intensity) is well known. Grassmann [1] proves this result directly by making use of a bound on the probability that all servers are busy. Independently, Lee and Cohen [2] derive this result by showing that the Erlang delay formula is a convex function. In this note, we provide a third method of proof, which exploits the relationship between the Erlang delay formula and the Poisson probability distribution. Several interesting intermediate results are also obtained.

A tandem Jackson network with feedback to the first node

AnN-node tandem queueing network with Bernoulli feedback to the end of the queue of thefirst node is considered. We first revisit the single-nodeM/G/1 queue with Bernoulli feedback, and derive a formula forEL(n), the expected queue length seen by a customer at his nth feedback. We show that, asn becomes large,EL(n) tends to ρ/(l ρ), ρ being the effective traffic intensity. We then treat the entire queueing network and calculate the mean value ofS, the total sojourn time of a customer in theN-node system. Based on these results we study the problem ofoptimally ordering the nodes so as to minimize ES. We show that this is a special case of a general sequencing problem and derive sufficient conditions for an optimal ordering. A few extensions of the serial queueing model are also analyzed. We conclude with an appendix in which we derive an explicit formula for the correlation coefficient between the number of customers seen by an arbitrary arrival to anM/G/1 queue, and the number of customers he leaves behind him upon departure. For theM/M/1 queue this coefficient simply equals the traffic intensity ρ.

Approximating queue lengths inM(t)/G/1 queue using the maximum entropy principle

Using the discrete time approach a model is developed for obtaining the expected queue length of theM(t)/G/1 queue. This type of queue occurs in different forms in transportation and traffic system...

Mathematical models for two kind of single-server queueing systems with arbitrary service-time distribution

We propose here a analytical method for the study of queueing systems with generalized time distributions. The idea is to define a new type of stochastic process, called a generalized Markov renewal process, which aay describe the transition behavior of the stochastic process at non-regenerative points. By means of the GMRP method, Mathematical models are then developed for two kind of typical single-server queueing systems which are the M/G/l/N/N and M/G/l/N queueing systems. The system performance Measures, such as the steady-state probability distribution of the number of customers in the system, expected queue length, expected number of customers in the system, average waiting time, mean busy period lengths, and so on, are derived for both the queueing systems.

Performances of integrated voice/data multiplexing system with controlled arrival

AbstractIn the integrated voice/data multiplexing system based on a time‐slotted frame format, the expected queue length of data packets increases significantly when the voice calls temporarily occupy all slots, even if the traffic loads of voice and data are light. To reduce such an increase of data queue, the integrated voice/data multiplexing system is investigated in this paper, and the arrival and service rates of voice calls are controlled with the number of slots occupied with voice calls. The system is analyzed exactly as an M/M/X queue by varying the number of servers, and the approximation method is presented for studying the behavior of a large‐number slot system. As a result, in a light data traffic load, it is effective in reducing the increase of data queue that the arrival and service rates of voice calls be controlled when a large number of slots are occupied by voice calls. However, in a heavy data traffic load, it is effective to use the movable boundary scheme in which necessary numbers of slots are reserved for data traffic.

On some queue length controlled stochastic processes

The authors study the input, output and queueing processes in a general controlled single‐server bulk queueing system. It is supposed that inter‐arrival time, service time, batch size of arriving units and the capacity of the server depend on the queue length.The authors establish an ergodicity criterion for both the queueing process with continuous time parameter and the embedded process, study their transient and steady state behavior and prove ergodic theorems for some functionals of the input, output and queueing processes. The following results are obtained: Invariant probability measure of the embedded process, stationary distribution of the process with continuous time parameter, expected value of a busy period, rates of input and output processes and the relative speed of convergence of the expected queue length. Various examples (including an optimization problem) illustrate methods developed in the paper.

An M/G/1 type vacation queuing model with exceptional service for certain customers

A queuing model is considered in which the management adopts a policy whereby the service counter will open if and only if at least a prescribed number, R + 1, of customers are available in each busy period. The additional waiting time for the first R customers causes the system to provide them with exceptional service composed of two stages. The other customers receive a possibly different service. All service times are assumed to follow general distributions, and customers arrive into the system according to a Poisson process where the parameter depends upon the class of states the system is in. Under these assumptions, partial generating functions for classes of state probabilities are derived, and expressions for the expected queue length, busy period, and waiting time are obtained. Further results are obtained in the special case where service times follow hyperexponential distributions.