Queueing with Cyclic Arrival Distributions and Variable Service Facilities: A Simulation Approach

This paper develops a near optimal policy for opening a second counter in a quick-service restaurant when n persons are in the first line, given that the arrival process is cyclic rather than purely stationary. A simulation was written in GPSS which permitted the use of different arrival generators during different time segments of the simulated day. Results indicate that as n is varied, waiting time and utilization levels exhibit an S-shaped curve between the pure 1 facility and the 2 facility cases. Results also indicate that a cyclic arrival process greatly increases (relative to the case of exponential arrivals) mean waiting time, facility utilization, and the range of n. Finding the optimal (least cost) value of n is illustrated for cases in which costs of customer waiting time can be meaningfully imputed.     

A two-class global FCFS discrete-time queueing model with arbitrary-length constant service times

We analyze a discrete-time queueing model where two types of customers, each having their own dedicated server, are accommodated in one single FCFS queue. Service times are deterministically equal to \(s \ge 1\) time slots each. New customers enter the system according to a general independent arrival process, but the types of consecutive customers may be nonindependent. As a result, arriving customers may (or may not) have the tendency to cluster according to their types, which may lead to more (or less) blocking of one type by the opposite type. The paper reveals the impact of this blocking phenomenon on the achievable throughput, the (average) system content, the (average) customer delay and the (average) unfinished work. The paper extends the results of earlier work where either the service times were assumed to be constant and equal to 1 slot each, or the customers all belonged to the same class. Our results show that, in case of Poisson arrivals, for given traffic intensity, the system-content distribution is insensitive to the length (s) of the service times, but the (mean) delay and the (mean) unfinished work in the system are not. In case of bursty arrivals, we find that all the performance measures are affected by the length (s) of the service times, for given traffic intensity.     

Class clustering destroys delay differentiation in priority queues

This paper considers a discrete-time priority queueing model with one server and two types (classes) of customers. Class-1 customers have absolute (service) priority over class-2 customers. New customer batches enter the system at the rate of one batch per slot, according to a general independent arrival process, i.e., the batch sizes (total numbers of arrivals) during consecutive time slots are i.i.d. random variables with arbitrary distribution. All customers entering the system during the same time slot (i.e., belonging to the same arrival batch) are of the same type, but customer types may change from slot to slot, i.e., from batch to batch. Specifically, the types of consecutive customer batches are correlated in a Markovian way, i.e., the probability that any batch of customers has type 1 or 2, respectively, depends on the type of the previous customer batch that has entered the system. Such an arrival model allows to vary not only the relative loads of both customer types in the arrival stream, but also the amount of correlation between the types of consecutive arrival batches. The results reveal that the amount of delay differentiation between the two customer classes that can be achieved by the priority mechanism strongly depends on the amount of such interclass correlation (or, class clustering) in the arrival stream. We believe that this phenomenon has been largely overlooked in the priority-scheduling literature.     

Log-Convexity of Counting Processes Evaluated at a Random end of Observation Time with Applications to Queueing Models

We consider a counting processes with independent inter-arrival times evaluated at a random end of observation time T, independent of the process. For instance, this situation can arise in a queueing model when we evaluate the number of arrivals after a random period which can depend on the process of service times. Provided that T has log-convex density, we give conditions for the inter-arrival times in the counting process so that the observed number of arrivals inherits this property. For exponential inter-arrival times (pure-birth processes) we provide necessary and sufficient conditions. As an application, we give conditions such that the stationary number of customers waiting in a queue is a log-convex random variable. We also study bounds in the approximation of log-convex discrete random variables by a geometric distribution.     

Analysis of queues in a random environment with impatient customers

We study M/M/c queues (c = 1, 1 < c < ∞ and c = ∞) in a Markovian environment with impatient customers. The arrivals and service rates are modulated by the underlying continuous-time Markov chain. When the external environment operates in phase 2, customers become impatient. We focus our attention on the explicit expressions of the performance measures. For each case of c, the corresponding probability generating function and mean queue size are obtained. Several special cases are studied and numerical experiments are presented.     

A Priority Discrete Queueing Model for Multimedia Multiplexers

Multiplexers have been extensively modeled as discrete time queueing systems. In this article, we model a multimedia multiplexer handling traffic of two classes. One class represents real-time traffic, e.g., packets of live audio or video transmissions, and the other nonreal-time traffic, e.g., packets of file transfer transmissions. These packets arrive into the multiplexer in batches. In each time slot, one batch of each class arrive. The multiplexer gives service priority to class-1 packets over class-2. The demands of each class are in conflict with that of the other, and thus they are treated by the multiplexer differently. The multiplexer is thus modeled as a (preemptive) priority discrete queueing system with simultaneous batch arrivals and geometric service time. The system occupancy is analyzed and the joint probability generating function (PGF) of the number of packets of each class is derived. From this PGF, marginal PGFs of interest are obtained. The results for deterministic service time, most suitable for ATM purposes, are readily obtainable as a special case from the results of this article.     

Further Results for the Queueing System <Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">x</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">c</Emphasis>

For the multi-channel bulk-arrival queue, M ^x /M/c, Abol'nikov and Kabak independently obtained steady state results. In this paper the results of these authors are extended, corrected and simplified. A number of measures of efficiency are calculated for three cases where the arrival group size has: (i) a constant value, (ii) a geometric distribution, or (iii) a positive Poisson distribution. The paper also shows how to calculate fractiles for both the queue length and the waiting time distribution. Examples of extensive numerical results for certain measures of efficiency are presented in tabular and chart form.     

Delay analysis of multiclass queues with correlated train arrivals and a hybrid priority/FIFO scheduling discipline

We analyze the delay experienced in a discrete-time priority queue with a train-arrival process. An infinite user population is considered. Each user occasionally sends packets in the form of trains: a variable number of fixed-length packets is generated and these packets arrive to the queue at the rate of one packet per slot. This is an adequate arrival process model for network traffic. Previous studies assumed two traffic classes, with one class getting priority over the other. We extend these studies to cope with a general number M of traffic classes that can be partitioned in an arbitrary number N of priority classes (1 ≤ N ≤ M). The lengths of the trains are traffic-class-dependent and generally distributed. To cope with the resulting general model, an (M × )∞-sized Markovian state vector is introduced. By using probability generating functions, moments and tail probabilities of the steady-state packet delays of all traffic classes are calculated. Since this study can be useful in deciding how to partition traffic classes in priority classes, we demonstrate the impact of this partitioning for some specific cases.     

On the Relative Waiting Times in the GI/M/s and the GI/M/1 Queueing Systems

The expected steady-state waiting time, Wq(s), in a GI/M/s system with interarrival-time distribution H(·) is compared with the mean waiting time, Wq, in an "equivalent" system comprised of s separate GI/M/1 queues each fed by an interarrival-time distribution G(·) with mean arrival rate equal to 1/s times that of H(·). For H(·) assumed to be Exponential, Gamma or Deterministic three possible relationships between H(·) and G(·) are considered: G(·) can be of the "same type" as H(·); G(·) can be derived from H(·) by assigning new arrivals to the individual channels in a cyclic order; and G(·) may be obtained from H(·) by assigning customers probabilistically to the different queues. The limiting behaviour of the ratio R = Wq/Wq(s) is studied for the extreme values (1 and 0) of the common traffic intensity, ρ. Closed form results, which depend on the forms of H(·) and G(·) and on the relationships between them, are derived. It is shown that Wq is greater than Wq(s) by a factor of at least (s + 1)/2 when ρ approaches one, and that R is at least s(s!) when ρ tends to zero. In the latter case, however, R goes to infinity (!) in most cases treated. The results may be used to evaluate the effect on the waiting times when, for certain (non-queueing) reasons, it is needed to partition a group of s servers into several small groups.     

Some related paradoxes of queuing theory: new cases and a unifying explanation

In this paper, we study a number of closely related paradoxes of queuing theory, each of which is based on the intuitive notion that the level of congestion in a queuing system should be directly related to the stochastic variability of the arrival process and the service times. In contrast to such an expectation, it has previously been shown that, in all H _k /G/1 queues, PW (the steady-state probability that a customer has to wait for service) decreases when the service-time becomes more variable. An analagous result has also been proved for p_loss (the steady-state probability that a customer is lost) in all H_k/G/1 loss systems. Such theoretical results can be seen, in this paper, to be part of a much broader scheme of paradoxical behaviour which covers a wide range of queuing systems. The main aim of this paper is to provide a unifying explanation for these kinds of behaviour. Using an analysis based on a simple, approximate model, we show that, for an arbitrary set of n GI/G_k/1 loss systems (k=1,..., n), if the interarrival-time distribution is fixed and ‘does not differ too greatly’ from the exponential distribution, and if the n systems are ordered in terms of their p_loss values, then the order that results whenever c_A<1 is the exact reverse of the order that results whenever c_A>1, where c_A is the coefficient of variation of the interarrival time. An important part of the analysis is the insensitivity of the p_loss value in an M/G/1 loss system to the choice of service-time distribution, for a given traffic intensity. The analysis is easily generalised to other queuing systems for which similar insensitivity results hold. Numerical results are presented for paradoxical behaviour of the following quantities in the steady state: p_loss in the GI/G/1 loss system; PW and W _q (the expected queuing time of customers) in the GI/G/1 queue; and p_K (the probability that all K machines are in the failed state) in the GI/G/r machine interference model. Two of these examples of paradoxical behaviour have not previously been reported in the literature. Additional cases are also discussed.     

A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue

We consider a model in which the production of new molecules in a chemical reaction network occurs in a seemingly stochastic fashion, and can be modeled as a Poisson process with a varying arrival rate: the rate is λ _i when an external Markov process J(?) is in state i. It is assumed that molecules decay after an exponential time with mean μ ^?1. The goal of this work is to analyze the distributional properties of the number of molecules in the system, under a specific time-scaling. In this scaling, the background process is sped up by a factor N ^α , for some α>0, whereas the arrival rates become N λ _i , for N large. The main result of this paper is a functional central limit theorem (F-CLT) for the number of molecules, in that, after centering and scaling, it converges to an Ornstein-Uhlenbeck process. An interesting dichotomy is observed: (i) if α > 1 the background process jumps faster than the arrival process, and consequently the arrival process behaves essentially as a (homogeneous) Poisson process, so that the scaling in the F-CLT is the usual \(\sqrt {N}\), whereas (ii) for α≤1 the background process is relatively slow, and the scaling in the F-CLT is N ^1?α/2. In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process J(?).     

Compound Geometric Distribution of Order <Emphasis Type="Italic">k</Emphasis>

The distribution of the number of trials until the first k consecutive successes in a sequence of Bernoulli trials with success probability p is known as geometric distribution of order k. Let T _k be a random variable that follows a geometric distribution of order k, and Y ₁,Y ₂,… a sequence of independent and identically distributed discrete random variables which are independent of T _k . In the present article we develop some results on the distribution of the compound random variable \(S_{k} =\sum_{t=1}^{T_{k}}Y_{t}\).     

Stationary analysis of the shortest queue problem

In this paper, we derive an approximation for throughput of TCP Compound connections under random losses. Throughput expressions for TCP Compound under a deterministic loss model exist in the literature. These are obtained assuming that the window sizes are continuous, i.e., a fluid behavior is assumed. We validate this model theoretically. We show that under the deterministic loss model, the TCP window evolution for TCP Compound is asymptotically periodic and is independent of the initial window size. We then consider the case when packets are lost randomly and independently of each other. We discuss Markov chain models to analyze performance of TCP in this scenario. We use insights from the deterministic loss model to get an appropriate scaling for the window size process and show that these scaled processes, indexed by p, the packet error rate, converge to a limit Markov chain process as p goes to 0. We show the existence and uniqueness of the stationary distribution for this limit process. Using the stationary distribution for the limit process, we obtain approximations for throughput, under random losses, for TCP Compound when packet error rates are small. We compare our results with ns2 simulations which show a good match and a better approximation than the fluid model at low p.     

Large deviations for the total queue size in non-Markovian tandem queues

We consider a d-node tandem queue with arrival process and light-tailed service processes at all queues i.i.d. and independent of each other. We consider three variations of the probability that the number of customers in the system reaches some high level N, namely during a busy cycle, in steady state, and upon arrival of a new customer. We show that their decay rates for large N have the same value and give an expression for this value.     

Analyzing <Emphasis Type="Italic">M/M/</Emphasis>1 Queues with Perturbations in the Arrival Process

This paper investigates when the M/M/1 model can be used to predict accurately the operating characteristics of queues with arrival processes that are slightly different from the Poisson process assumed in the model. The arrival processes considered here are perturbed Poisson processes. The perturbations are deviations from the exponential distribution of the inter-arrival times or from the assumption of independence between successive inter-arrival times. An estimate is derived for the difference between the expected numbers in perturbed and M/M/1 queueing systems with the same traffic intensity. The results, for example, indicate that the M/M/1 model can predict the performance of the queue when the arrival process is perturbed by inserting a few short inter-arrival times, an occasional batch arrival or small dependencies between successive inter-arrival times. In contrast, the M/M/1 is not a good model when the arrival process is perturbed by inserting a few long inter-arrival times.     

Batch queues,reversibility and first-passage percolation

We consider a model of queues in discrete time, with batch services and arrivals. The case where arrival and service batches both have Bernoulli distributions corresponds to a discrete-time M/M/1 queue, and the case where both have geometric distributions has also been previously studied. We describe a common extension to a more general class where the batches are the product of a Bernoulli and a geometric, and use reversibility arguments to prove versions of Burke’s theorem for these models. Extensions to models with continuous time or continuous workload are also described. As an application, we show how these results can be combined with methods of Seppäläinen and O’Connell to provide exact solutions for a new class of first-passage percolation problems.     

Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation

In this paper, a piecewise constant time-stepping discontinuous Galerkin method combined with a piecewise linear finite element method is applied to solve control constrained optimal control problem governed by time fractional diffusion equation. The control variable is approximated by variational discretization approach. The discrete first-order optimality condition is derived based on the first discretize then optimize approach. We demonstrate the commutativity of discretization and optimization for the time-stepping discontinuous Galerkin discretization. Since the state variable and the adjoint state variable in general have weak singularity near t =?0and t = T, a time adaptive algorithm is developed based on step doubling technique, which can be used to guide the time mesh refinement. Numerical examples are given to illustrate the theoretical findings.     

Univariate Stochastic Programming with Random Decision Variable

The problem of minimising E(X) subject to the constraints X ? 0, P(X ? b) ? a(0 < a < 1) has been considered, where b is a non-negative random variable with continuous probability distribution. A necessary and sufficient condition for randomised decisions to be superior to the non-randomised one has been derived.     

Markov decision processes with noise-corrupted and delayed state observations

We consider the partially observed Markov decision process with observations delayed by k time periods. We show that at stage t, a sufficient statistic is the probability distribution of the underlying system state at stage t - k and all actions taken from stage t - k through stage t - 1. We show that improved observation quality and/or reduced data delay will not decrease the optimal expected total discounted reward, and we explore the optimality conditions for three important special cases. We present a measure of the marginal value of receiving state observations delayed by (k - 1) stages rather than delayed by k stages. We show that in the limit as k →∞ the problem is equivalent to the completely unobserved case. We present numerical examples which illustrate the value of receiving state information delayed by k stages.     

Structural properties and stochastic bounds for a buffer problem in packetized voice transmission

We consider a communication channel which carries packetized voice. A fixed number (K) of calls are being transmitted. Each of these calls generates one packet at everyC timeslots and the channel can transmit at most one packet every timeslot. We consider the nontrivial caseK ≤C. We study the effectsK, C and the arrival process have on the number of packets in the buffer. When the call origination epochs in the firstC timeslots of theK calls are uniformly distributed (i.e. when the arrivals during the firstC timeslots have a multinomial distribution) it is shown that the stationary number of calls waiting in the buffer is stochastically increasing and convex in the number of calls. For a fixed average number of calls per slot, it is shown that increasing the number of slots per frame increases the stationary number of packets in the buffer in the sense of increasing convex ordering. Using this, it is shown that the stationary number of packets in the buffer is bounded from above by the number of packets in a stationary discreteM/D/1 queue with arrival rateK/C and unit service time. This bound is in the sense of the increasing convex order.