External Arrival Processes Research Articles

Queing Theory, a Tool for Polio Eradication in Nigeria

To curb the spread of contagious diseases and the recent polio outbreak in Nigeria, health departments must set up and operate clinics to dispense medications or vaccines. Residents arrive according to an external (not necessarily Poisson) Arrival process to the clinic. When a resident arrives, he goes to the first workstation, based on his or her information, the resident moves from one workstation to another in the clinic. The queuing network is decomposed by estimating the performance of each workstation using a combination of exact and approximate models. A key contribution of this research is to introduce approximations for workstations with batch arrivals and multiple parallel servers, for workstations with batch service processes and multiple parallel servers, and for self service workstations. We validated the models for likely scenarios using data collected from one of the states vaccination clinics in the country during the vaccination exercises.

How Wireless Queues Benefit from Motion: An Analysis of the Continuum Between Zero and Infinite Mobility

This paper considers the time evolution of a queue that is embedded in a Poisson point process of moving wireless interferers. The queue is driven by an external arrival process and is subject to a time-varying service process that is a function of the SINR that it sees. Static configurations of interferers result in an infinite queue workload with positive probability. In contrast, a generic stability condition is established for the queue in the case where interferers possess any non-zero mobility that results in displacements that are both independent across interferers and oblivious to interferer positions. The proof leverages the mixing property of the Poisson point process. The effect of an increase in mobility on queuing metrics is also studied. Convex ordering tools are used to establish that faster moving interferers result in a queue workload that is smaller for the increasing-convex stochastic order. As a corollary, mean workload and mean delay decrease as network mobility increases. This stochastic ordering as a function of mobility is explained by establishing positive correlations between SINR level-crossing events at different time points, and by determining the autocorrelation function for interference and observing that it decreases with increasing mobility. System behaviour is empirically analyzed using discrete-event simulation and the performance of various mobility models is evaluated using heavy-traffic approximations.

A robust queueing network analyzer based on indices of dispersion

AbstractWe develop a robust queueing network analyzer algorithm to approximate the steady‐state performance of a single‐class open queueing network of single‐server queues with Markovian routing. The algorithm allows nonrenewal external arrival processes, general service‐time distributions and customer feedback. The algorithm is based on a decomposition approximation, where each flow is partially characterized by its rate and a continuous function that measures the stochastic variability over time. This function is a scaled version of the variance‐time curve, called the index of dispersion for counts (IDC). The required IDC functions for the external arrival processes can be calculated from the model primitives or estimated from data. Approximations for the IDC functions of the internal flows are calculated by solving a set of linear equations. The theoretical basis is provided by heavy‐traffic limits for the flows established in our previous papers. A robust queueing technique is used to generate approximations of the mean steady‐state performance at each queue from the IDC of the total arrival flow and the service specification at that queue. The algorithm's effectiveness is supported by extensive simulation studies.

Queue-length balance equations in multiclass multiserver queues and their generalizations

A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for multidimensional queue-length processes for a quite general network of multiclass multiserver queues. We formally derive those balance equations under a general framework. They are called distributional relationships and are obtained for any external arrival process and state-dependent routing as long as certain stationarity conditions are satisfied and external arrivals and service completions do not simultaneously occur. We demonstrate the use of these balance equations, in combination with PASTA, by (1) providing very simple derivations of some known results for polling systems and (2) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a nonstationary framework.

Nonparametric Inference for Queueing Networks of GeomX/G/∞ Queues in Discrete Time

We study nonparametric estimation problems for discrete-time stochastic networks of GeomX/G/∞ queues. We assume that we are only able to observe the external arrival and external departure processes at the nodes over a stretch of time. Based on such incomplete information of the system, we aim to construct estimators for the unknown general service time distributions at the nodes without imposing any parametric condition. We propose two different estimation approaches. The first approach is based on the construction of a so-called sequence of differences, and a crucial relation between the expected number of external departures at a node and specific sojourn time distributions in the network. The second approach directly utilizes the structure of the cross-covariance functions between external arrival and departure processes at the nodes. Both methods lead to deconvolution problems which we solve explicitly. A detailed simulation study illustrates the numerical performances of our estimators and shows their advantages and disadvantages.

Nonparametric Inference for Queueing Networks of GeomX/G/∞ Queues in Discrete Time

We study nonparametric estimation problems for discrete-time stochastic networks of Geom X /G/∞ queues. We assume that we are only able to observe the external arrival and external departure processes at the nodes over a stretch of time. Based on such incomplete information of the system, we aim to construct estimators for the unknown general service time distributions at the nodes without imposing any parametric condition. We propose two different estimation approaches. The first approach is based on the construction of a so-called sequence of differences, and a crucial relation between the expected number of external departures at a node and specific sojourn time distributions in the network. The second approach directly utilizes the structure of the cross-covariance functions between external arrival and departure processes at the nodes. Both methods lead to deconvolution problems which we solve explicitly. A detailed simulation study illustrates the numerical performances of our estimators and shows their advantages and disadvantages.

Nonparametric inference for stochastic feedforward networks based on cross-spectral analysis of point processes

In this paper we study a nonparametric estimation problem for stochastic feedforward systems with nodes of type G/G/$\infty.$ We assume that we have observations of the external arrival and external departure processes of customers of the system, but no information about the movements of the indistinguishable customers in the network. Our aim is the construction of estimators for the characteristic functions and the densities of the service time distributions at the nodes as well as for the routing probabilities. Since the system is only partly observed we have to clarify first if the parameters are identifiable from the given data. The crucial point in our approach is the observation that in our stochastic networks under study the influence of the arrival processes on the departure processes can be described in a linear and time-invariant model. This makes it possible to apply cross-spectral techniques for multivariate point processes. The construction of the estimators is then based on smoothed periodograms. We prove asymptotic normality for the estimators. We present the statistical analysis for a tandem system of two nodes in full details and show afterwards how the results can be generalized to feedforward systems of three or more nodes and to systems with positive feedback probabilities at the nodes.

On the large deviations behavior of acyclic networks of $G/G/1$ queues

We consider a single class, acyclic network of $G/G/1$ queues. We impose some mild assumptions on the service and external arrival processes and we characterize the large deviations behavior of all the processes resulting from various operations in the network. For the network model that we are considering, these operations are passing-through-a-single-server-queue (the process resulting from this operation being the departure process), superposition of independent processes and deterministic splitting of a process into a number of processes. We also characterize the large deviations behavior of the waiting time and the queue length observed by a typical customer in a single server queue. We prove that the assumptions imposed on the external arrival processes are preserved by these operations, and we show how to apply inductively these results to obtain the large deviations behavior of the waiting time and the queue length in all the queues of the network. Our results indicate how these large deviations occur, by concretely characterizing the most likely path that leads to them.

Bounds on the Mean Delay in Multiclass Queueing Networks under Shortfall-Based Priority Rules

A significant amount of recent research has been focused on the stability of multiclass open networks of queues (MONQs). It has been shown that these networks may be unstable under various queueing disciplines even when at each one of the nodes the arrival rate is less than the service rate. Clearly, in such cases the expected delay and the expected number of customers in the system are infinite. In this paper we propose a new class of scheduling rules that can be used in multiclass queueing networks. We refer to this class as the stable shortfall-based priority (SSBP) rules. This SSBP class itself belongs to a larger class of rules, which we refer to as the shortfall-based priority (SBP) rules. SBP is a generalization of the standard non-preemptive priority rule in which customers of the same priority class are served first-come, first-served (FCFS). Rules from SBP can mimic FCFS as well as the so-called strict or head-of-the-line priority disciplines. We show that the use of any rule from the SSBP class ensures stability in a broad class of MONQs found in practice. We proceed with the construction of a sample path inequality for the work done by an SSBP server and show how this inequality can be used to derive upper bounds for the delay when service times are bounded. Bounds for the expected delay of each class of customers in an MONQ are then obtained under the assumptions that the external arrival processes have i.i.d. interarrival times, the routings are deterministic and the service times at each step of the route are bounded. In order to derive these bounds for the average delay in an MONQ we make use of some of the classical ideas of Kingman.

Variability Functions for Parametric-Decomposition Approximations of Queueing Networks

We propose an enhancement to the parametric-decomposition method for calculating approximate steady-state performance measures of open queueing networks with non-Poisson arrival processes and nonexponential service-time distributions. Instead of using a variability parameter ca2 for each arrival process, we suggest using a variability function ca2ρ, 0 < ρ < 1, for each arrival process i.e, the variability parameter should be regarded as a function of the traffic intensity ρ of a queue to which the arrival process might go. Variability functions provide a convenient representation of different levels of variability in different time scales for arrival processes that are not nearly renewal processes. Variability functions enable the approximations to account for long-range effects in queueing networks that cannot be addressed by variability parameters. For example, the variability functions provide a way to address the heavy-traffic bottleneck phenomenon, in which exceptional variability (either high or low) in the input has little impact in a series of queues with low-to-moderate traffic intensities, and then has a big impact when it reaches a later queue with a relatively high traffic intensity. The variability functions also enable the approximations to characterize irregular periodic deterministic external arrival processes in a reasonable way; i.e., if there are no batches, then ca2ρ should be 0 for ρ near 0 or 1, but ca2ρ can assume arbitrarily large values for appropriate intermediate ρ. We present a full network algorithm with variability functions, showing that the idea is implementable. We also show how simulations of single queues can be effectively exploited to determine variability functions for difficult external arrival processes.

Analysis of a tandem queue with state-dependent general blocking: A GSMP perspective

In this paper, we study a tandem queue where there is a finite number of buffer positions at each stage. The blocking scheme is general in the sense that it can model a number of classical blocking schemes, including communication, manufacturing and kanban blockings as special cases. The system considered here differs from the conventional system in two aspects: (1) departure of jobs from the system is determined by an external arrival process of another queue lying parallel to the tandem queue; (2) the control parameters of the blocking scheme are state-dependent in that they may change values depending on the state of the system. We model this system as a generalized semi-Markov process (GSMP), we study the structural properties of its scheme, and establish monotonicity and convexity properties of event times with respect to both the clock times and the integer blocking parameters. In particular, we demonstrate that the state-dependent scheme is “better”, in certain sense, than the corresponding static scheme. Our results also recover the structural properties previously established for the classical blockings.

Comparisons of multi-server queues with finite waiting rooms

In this paper we consider s-server queues with capacity c, 1 ≤ s ≤ c ≤ ∞, the first-come first-served queue discipline and very general arrival and service processes. We show that the admission epochs and departure epochs decrease, so that the throughput increases, when any of the following changes occur: (1) the number of servers s increases, (2) the capacity c increases, (3) the external arrival counting process increases or (4) the service times decrease, provided that the service times are assigned in order of service initiation and that a subsequence ordering is used to compare arrival counting processes. The subsequence ordering for the arrival processes is very important for obtaining positive results with finite waiting rooms. The subsequence ordering holds between a superposition point process and its component point processes. The subsequence ordering can often be applied via its stochastic generalization, the stochastic subsequence ordering, which is implied by a failure rate ordering.

Arranging Queues in Series: A Simulation Experiment

For given external arrival process and given service-time distributions, the object is to determine the order of infinite-capacity single-server queues in series that minimizes the long-run average sojourn time per customer. We gain additional insight into this queueing design problem, and congestion in non-Markov open queueing networks more generally, by performing simulations for the case of two queues. For this design problem, we conclude that the key issue is variability: The order tends to matter more when the service-time distributions have significantly different variability, and less otherwise. Arranging the queues in order of increasing service-time variability, using the squared coefficient of variation as a partial characterization of variability, seems to be an effective simple design heuristic. Parametric-decomposition approximations seem to provide relatively good quantitative estimates of how much the order matters.

A Light-Traffic Approximation for Single-Class Departure Processes from Multi-Class Queues

This paper discusses an approximation for single-class departure processes from multi-class queues: If the arrival rate of one class upon one visit to the queue is a small proportion of the total arrival rate there, then the departure process for that class from that visit should be nearly the same as the arrival process for that class for that visit. This can be regarded as a light-traffic approximation, but only the one class must be in light traffic; the overall traffic intensity of the queue need not be low. As a consequence, in a queueing network if the routing for one class is deterministic, and if the light-traffic condition applies at every queue this class visits, then the arrival and departure processes for this class at each visit to each queue should be nearly the same as its external arrival process. This approximation is explained in terms of different time scales, and is justified here by a limit theorem in a special case. There are important implications for parametric decomposition approximation techniques: the variability parameter partially characterizing the departure process at any visit to any queue of such a low-intensity class should be nearly the same as the variability parameter partially characterizing the arrival process for that class at that visit to that queue. The approximation principle in this form was recently proposed by G. Bitran and D. Tirupati while developing improved parametric-decomposition approximations for low-variability multi-class queueing networks with deterministic routing, which have important, applications in manufacturing. The approximation principle also has important implications for data networks showing how burstiness in originated traffic can pass through heavily shared network facilities where it has relatively little effect and then reappear at the destination.

Approximation for the departure process of a queue in a network

AbstractA simple renewal process is identified to approximate the complex departure process of a queue often found in queueing network models. The arrival process to the queue is the superposition or merging of several independent component‐renewal processes that are approximations of departure processes from other queues and external arrival processes; there is a single server with exponential service times, and the waiting space is infinite. The departure process of this queue is of interest because it is the arrival process to other queues in the network. The approximation proposed is a hybrid; the mean and variance of the approximating departure intervals is a weighted average of those determined by basic methods in Whitt [41] with the weighting function empirically determined using simulation. Tandem queueing systems with superposition arrival processes and exponential service times are used to evaluate the approximation. The departure process of the first queue in the tandem is approximated by a renewal process, the tandem system is replaced by two independent queues, and the second queue is solved analytically. When compared to simulation estimates, the average absolute error in hybrid approximations of the expected number in the second queue is 6%, a significant improvement over 22–41% in the basic methods.

The Best Order for Queues in Series

An important design problem for queueing systems is to determine the best order for two or more service stations in series. For given external arrival process and given service-time distributions, the object is to determine the order of the stations (to be used by all customers) that minimizes the expected equilibrium sojourn time per customer. Unfortunately, very little is known about this problem because exact analysis is extremely difficult. This paper applies approximation methods recently developed for networks of queues to obtain approximation formulas and useful heuristic design principles.

The Queueing Network Analyzer

This paper describes the Queueing Network Analyzer (QNA), a software package developed at Bell Laboratories to calculate approximate congestion measures for a network of queues. The first version of QNA analyzes open networks of multiserver nodes with the first-come, first-served discipline and no capacity constraints. An important feature is that the external arrival processes need not be Poisson and the service-time distributions need not be exponential. Treating other kinds of variability is important. For example, with packet-switched communication networks we need to describe the congestion resulting from bursty traffic and the nearly constant service times of packets. The general approach in QNA is to approximately characterize the arrival processes by two or three parameters and then analyze the individual nodes separately. The first version of QNA uses two parameters to characterize the arrival processes and service times, one to describe the rate and the other to describe the variability. The nodes are then analyzed as standard GI/G/m queues partially characterized by the first two moments of the interarrival-time and service-time distributions. Congestion measures for the network as a whole are obtained by assuming as an approximation that the nodes are stochastically independent given the approximate flow parameters.