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In complex systems, it is quite common to resort to approximations when optimizing system performance. These approximations typically involve selecting a particular system parameter and then studying the performance of the system as this parameter grows without bound. Indexing this parameter by n, we prove that if the approximation to the objective function is accurate up to O(f(n)) for some function f(n), then under some regularity conditions, the prescriptions that are derived from this approximation are in fact o(f(n))-optimal. A consequence of this result is that the well-known square-root staffing rules for capacity sizing in M/M/N and M/M/N+M queues to minimize the sum of linear expected steady-state customer waiting costs and linear capacity costs are o(1)-optimal, i.e., their optimality gap is asymptotically zero.

This paper studies a scheduling control problem for a single-server multiclass queueing network in heavy traffic, operating in a changing environment. The changing environment is modeled as a finite state Markov process that modulates the arrival and service rates in the system. Various cases are considered: fast changing environment, fixed environment and slow changing environment. In each of the cases, using weak convergence analysis, in particular functional limit theorems for renewal processes and ergodic Markov processes, it is shown that an appropriate "averaged" version of the classical c\mu -policy (the priority policy that favors classes with higher values of the product of holding cost c and service rate \mu) is asymptotically optimal for an infinite horizon discounted cost criterion.

We study the scheduling polices for asymptotically optimal delay in queueing systems with switching overhead. Such systems consist of a single server that serves multiple queues, and some capacity is lost whenever the server switches to serve a different set of queues. The capacity loss due to this switching overhead can be significant in many emerging applications, and needs to be explicitly addressed in the design of scheduling policies. For example, in 60GHz wireless networks with directional antennas, base stations need to train and reconfigure their beam patterns whenever they switch from one client to another. Considerable switching overhead can also be observed in many other queueing systems such as transportation networks and manufacturing systems. While the celebrated Max-Weight policy achieves asymptotically optimal average delay for systems without switching overhead, it fails to preserve throughput-optimality, let alone delay-optimality, when switching overhead is taken into account. We propose a class of Biased Max-Weight scheduling policies that explicitly takes switching overhead into account. The Biased Max-Weight policy can use either queue length or head-of-line waiting time as an indicator of the system status. We prove that our policies not only are throughput-optimal, but also can be made arbitrarily close to the asymptotic lower bound on average delay. To validate the performance of the proposed policies, we provide extensive simulation with various system topologies and different traffic patterns. We show that the proposed policies indeed achieve much better delay performance than that of the state-of-the-art policy.

During the last decade bike sharing systems have emerged as a public transport mode in urban short trips in more than 500 major cities around the world. For the mobility service mode, many challenges from its operations are not well addressed yet, for example, how to develop the bike sharing systems to be able to effectively satisfy the fluctuating demands both for bikes and for vacant lockers. To this end, it is a key to give performance analysis of the bike sharing systems. This paper first describes a large-scale bike sharing system. Then the bike sharing system is abstracted as a closed queueing network with multi-class customers, where the virtual customers and the virtual nodes are set up, and the service rates as well as the relative arrival rates are established. Finally, this paper gives a product-form solution to the steady state joint probabilities of queue lengths, and gives performance analysis of the bike sharing system. Therefore, this paper provides a unified framework for analyzing closed queueing networks in the study of bike sharing systems. We hope the methodology and results of this paper can be applicable in the study of more general bike sharing systems.

We study the stability of single server retrial queues under general distribution for retrial times and stationary ergodic service times, for three main retrial policies studied in the literature: classical linear, constant and control policies. The approach used is the renovating events approach to obtain sufficient stability conditions by strong coupling convergence of the process modeling the dynamics of the system to a unique stationary ergodic regime. We also obtain instability conditions by convergence in distribution to improper limiting sequences.

A methodology is presented to compute performance measures for distributed information systems with unbalanced flows (i.e., number of transaction leaving a server is not the same as number of transactions entering that server) due to asynchronously spawned parallel tasks--an increasingly important phenomenon in modern information systems which has a significant effect on performance that cannot be analyzed by classical queueing network models. A decomposition method is applied to decompose the unbalanced flows. Formulaw for open queueing networks with unbalanced flows due to asynchronously spawned tasks are developed. Furthermore, An algorithm based on Buzen's convolution algorithm is developed to test the necessary and sufficient condition for closed system stability as well as to compute performance measures. An average of less than four iterations is necessary for convergence with this algorithm. A study of the INFOPLEX data storage hierarchy has been conducted using this rapid solution algorithm and detailed simulations; highly consistent results were obtained. A cost effective software tool, using this methodology, has been developed to analyze an architectural design and to produce measures such as throughput, utilization, and response time so that the potential performance problems can be identified. (Author)

In modern computer systems, jobs are divided into short tasks and executed in parallel. Empirical observations in practical systems suggest that the task service times are highly random and the job service time is bottlenecked by the slowest straggling task. One common solution for straggler mitigation is to replicate a task on multiple servers and wait for one replica of the task to finish early. The delay performance of replications depends heavily on the scheduling decisions of when to replicate, which servers to replicate on, and which job to serve first. So far, little is understood on how to optimize these scheduling decisions for minimizing the delay to complete the jobs. In this paper, we present a comprehensive study on delay-optimal scheduling of replications in both centralized and distributed multi-server systems. Low-complexity scheduling policies are designed and are proven to be delay-optimal or near delay-optimal in stochastic ordering among all causal and non-preemptive policies. These theoretical results are established for general system settings and delay metrics that allow for arbitrary arrival processes, arbitrary job sizes, arbitrary due times, and heterogeneous servers with data locality constraints. Novel sample-path tools are developed to prove these results.

The problem of controlling M/M/c queueing systems with c = or 1 is considered. By providing a new definition of the time of transition, we enlarge the standard set of decision epochs, and obtain a preferred version of the n- period problem in which the times between transitions are exponential random variables with constant parameter. Using this new technique, we are able to use the inductive approach in a manner characteristic of inventory theory. The efficacy of the approach is then demonstrated by successfully finding the form of an optimal policy for four quite distinct models that have appeared in the literature; namely, those of (i) McGill, (ii) Miller-Cramer, (iii) Crabill-Sabeti, and (iv) Low. Of particular note, one analysis establishes that an (s, S) or control-limit policy is optimal for an M/M/c queue with switching costs and removable servers.

The interest to retrial queueing systems is due to their application to telephone systems. The paper studies multiserver retrial queueing systems with $n$ servers. Arrival process is a quite general point process. An arriving customer occupies one of free servers. If upon arrival all servers are busy, then the customer waits for his service in orbit, and after random time retries more and more to occupy a server. The orbit has one waiting space only, and arriving customer, who finds all servers busy and the waiting space occupied, losses from the system. Time intervals between possible retrials are assumed to have arbitrary distribution (the retrial scheme is exactly explained in the paper). The paper provides analysis of this system. Specifically the paper studies optimal number of servers to decrease the loss proportion to a given value. The representation obtained for loss proportion enables us to solve the problem numerically. The algorithm for numerical solution includes effective simulation, which meets the challenge of rare events problem in simulation.

We consider optimizing average queueing delay and average power consumption in a nonpreemptive multi-class M/G/1 queue with dynamic power control that affects instantaneous service rates. Four problems are studied: (1) satisfying per-class average delay constraints; (2) minimizing a separable convex function of average delays subject to per-class delay constraints; (3) minimizing average power consumption subject to per-class delay constraints; (4) minimizing a separable convex function of average delays subject to an average power constraint. Combining an achievable region approach in queueing systems and the Lyapunov optimization theory suitable for optimizing dynamic systems with time average constraints, we propose a unified framework to solve the above problems. The solutions are variants of dynamic $c\mu$ rules, and implement weighted priority policies in every busy period, where weights are determined by past queueing delays in all job classes. Our solutions require limited statistical knowledge of arrivals and service times, and no statistical knowledge is needed in the first problem. Overall, we provide a new set of tools for stochastic optimization and control over multi-class queueing systems with time average constraints.

An overview of the recursive equations based models and their applications in simulation based analysis and optimization of queueing systems is given. These models provide a variety of systems with a convenient and unified representation in terms of recursions for arrival and departure times of customers, which involves only the operations of maximum, minimum, and addition.

The dynamics of pedestrian crowds has been studied intensively in recent years, both theoretically and empirically. However, in many situations pedestrian crowds are rather static, e.g. due to jamming near bottlenecks or queueing at ticket counters or supermarket checkouts. Classically such queues are often described by the M/M/1 queue that neglects the internal structure (density profile) of the queue by focussing on the system length as the only dynamical variable. This is different in the Exclusive Queueing Process (EQP) in which the queue is considered on a microscopic level. It is equivalent to a Totally Asymmetric Exclusion Process (TASEP) of varying length. The EQP has a surprisingly rich phase diagram with respect to the arrival probability alpha and the service probability beta. The behavior on the phase transition line is much more complex than for the TASEP with a fixed system length. It is nonuniversal and depends strongly on the update procedure used. In this article, we review the main properties of the EQP. We also mention extensions and applications of the EQP and some related models.

Multiuser Multi-Packet Transmission (MPT) from an Access Point (AP) equipped with multiple antennas to multiple single antenna nodes can be achieved by exploiting the spatial dimension of the channel. In this paper we present a queueing model to analytically study such systems from the link-layer perspective, in presence of random packet arrivals and heterogeneous channel conditions. The analysis relies on a blind estimation of the number of different destinations among the packets waiting in the queue, which allows for building a simple, but general model for MPT systems. Simulation results validate the accuracy of the analytical model and provide further insights on the cross-relations between the channel state, the number of antennas, and the number of active users, as well as how they affect the system performance. The simplicity and accuracy of the model makes it suitable for the evaluation of link-layer protocols supporting multiuser MPT in non-saturation conditions, where the queueing dynamics play an important role on the achieved performance.

In most queueing situations, it is desirable that service to customers be free from interruptions causing time and service losses. It is recognized, however, that in certain circumstances controlled interruptions may improve overall system performance, and this idea is the bais of all time- sharing systems. In the latter, service is given in segments; the customer may be served, uninterrupted, for no longer than some predetermined time interval called a quantum. If the service is not completed within a given quantum, the customer is dismissed and placed in a queue, and the next customer is admitted. This paper is a survey of time-sharing models studied recently by the authors. It covers the following single-server models: (1) single queue with infinite number of potential customers - R.R.1; (2) single queue with finite number of potential customers; (3) r queues with infinite number of potential customers - R.R.r, in which a customer who completes his i-th service segment joins the end of the (i+1)-th queue, the r-th queue is organized on a 'round-robin' basis, and the server, when admitting a customer to service, selects the first in the lowest-index's non-empty queue; (4) R.R.1. with various types of priority regimes. Each model provides a means for achieving desired given properties. The performance parameters of the models are compared numerically, and the advantages and weaknesses of each model are discussed.

In most queueing situations, it is desirable that service to customers be free from interruptions causing time and service losses. It is recognized, however, that in certain circumstances controlled interruptions may improve overall system performance, and this idea is the bais of all time- sharing systems. In the latter, service is given in segments; the customer may be served, uninterrupted, for no longer than some predetermined time interval called a quantum. If the service is not completed within a given quantum, the customer is dismissed and placed in a queue, and the next customer is admitted. This paper is a survey of time-sharing models studied recently by the authors. It covers the following single-server models: (1) single queue with infinite number of potential customers - R.R.1; (2) single queue with finite number of potential customers; (3) r queues with infinite number of potential customers - R.R.r, in which a customer who completes his i-th service segment joins the end of the (i+1)-th queue, the r-th queue is organized on a 'round-robin' basis, and the server, when admitting a customer to service, selects the first in the lowest-index's non-empty queue; (4) R.R.1. with various types of priority regimes. Each model provides a means for achieving desired given properties. The performance parameters of the models are compared numerically, and the advantages and weaknesses of each model are discussed.

A two node tandem queueing system with phase-type servers and Bernoulli arrivals is considered in discrete-time when servers are subject to blocking and failures. The invanant probability vector of the the underlying finite state Quasi-Birth-and-Death process is shown to admit a matrix- geometric representation for all values of the arrival rate wavelength. The corresponding rate matrix is given explicitly in terms of the modA parameters and the resulting closed-form expression provides the basis for an efficient calculation of the invariant probability vector. The cases wavelength = 1 and wavelength 1 are studied separately and the irreducibilty of the underlying Markov chain is discussed for each case. The continuous-time formulation is bnefly discussed and only major differences with the discrete-time results are pointed out. Some numerical examples are also provided.

The paper discusses the literature on application of the methods of queueing theory to the analysis of information systems. The principles of queueing theory are given, as are the most widely used results as applied to the simplest queueing systems. Priority systems are discussed and a simple example of their use is given. A survey of models of multi-access systems with time sharing is given formulated in the language of queueing theory.

In a packet switched network, several types of transmission may connect a pair of nodes of the network. The delay characteristics of each type of transmission may be significantly different and so a single queue to the node pair operating under a first-come, first-served rule may not be best. This paper develops both exact and approximate mathematical models that determine various system-level performance measures for a (K,N) scheme. Under this scheme arriving packets are placed in the queue for one type of transmission until the size reaches K, whereupon arriving packets are diverted to the other transmission system. When the number of packets at the first system drops to N(K), arriving packets are placed in the first queue.

In a packet switched network, several types of transmission may connect a pair of nodes of the network. The delay characteristics of each type of transmission may be significantly different and so a single queue to the node pair operating under a first-come, first-served rule may not be best. This paper develops both exact and approximate mathematical models that determine various system-level performance measures for a (K,N) scheme. Under this scheme arriving packets are placed in the queue for one type of transmission until the size reaches K, whereupon arriving packets are diverted to the other transmission system. When the number of packets at the first system drops to N(K), arriving packets are placed in the first queue.

In a packet switched network, several types of transmission may connect a pair of nodes of the network. The delay characteristics of each type of transmission may be significantly different and so a single queue to the node pair operating under a first-come, first-served rule may not be best. This paper develops both exact and approximate mathematical models that determine various system-level performance measures for a (K,N) scheme. Under this scheme arriving packets are placed in the queue for one type of transmission until the size reaches K, whereupon arriving packets are diverted to the other transmission system. When the number of packets at the first system drops to N(K), arriving packets are placed in the first queue.

Includes bibliographical references (p. 32-34)

This paper analyzes a fault-tolerant computer system. The failure/repair behavior of the system is modeled by an irreducible continuous-time Markov chain. Jobs arrive in a Poisson fashion to the system and are serviced according to an FCFS discipline. A failure may cause the loss of the work already done on the job in service, if any; in this case the interrupted job is repeated as soon as the system is ready to deliver service. In addition to the delays due to failures and repairs, jobs suffer delays due to queueing. The authors presents a queueing analysis of fault-tolerant systems and study the steady-state behavior of the number of jobs in the system. As a numerical example, they consider a system with two processors subject to failures and repairs. (Author)

Operational analysis replaces certain classical queueing theory assumptions with the conditions of 'homogeneous service times' and 'on-line=off-line behavior.' In the general case, it has been conjectured that these conditions hold as t--infinity only if the service times are exponentially distributed. In this paper, we show that this is correct for stable M/G/1 queueing systems. We also state dual results for inter-arrival times in G/M/1. Finally, we consider the relationship between the operational quantities S(n) and the mean service time in M/G/1. This relationship is shown to depend on the form of the service time distribution. It follows that using operational analysis to predict the performance of an M/G/1 queueing system will be most successful when the service time is exponential. Simulation evidence is presented which supports this claim. (Author)

This dissertation contians studies in two related areas: Discrete Event Systems Theory and Queueing Network Theory. In the first line of study, deterministic discrete event systems are modeled by formal automata- like structures, and a hierarchy of morphic relations among them is developed. A canonical representation of stochastic discrete event systems in coordinate probability space is proposed, and a hierarchy of morphic relations among them is constructed by means of measure preserving transformations. In the second line of study, several operating characteristics of the class of Jackson queueing networks are investiaged. Included are: line sizes (network state), total service awarded to customers, and traffic processes on the arcs. Special emphasis is placed on rigorous derivations of results from solid mathematical and statistical foundations. In the process, a number of theoretical gaps in the extant theory of state equilibrium are closed, and Burke's Theorem is extended from M/M/1 queues to Jackson networks with single server nodes.

This dissertation contians studies in two related areas: Discrete Event Systems Theory and Queueing Network Theory. In the first line of study, deterministic discrete event systems are modeled by formal automata- like structures, and a hierarchy of morphic relations among them is developed. A canonical representation of stochastic discrete event systems in coordinate probability space is proposed, and a hierarchy of morphic relations among them is constructed by means of measure preserving transformations. In the second line of study, several operating characteristics of the class of Jackson queueing networks are investiaged. Included are: line sizes (network state), total service awarded to customers, and traffic processes on the arcs. Special emphasis is placed on rigorous derivations of results from solid mathematical and statistical foundations. In the process, a number of theoretical gaps in the extant theory of state equilibrium are closed, and Burke's Theorem is extended from M/M/1 queues to Jackson networks with single server nodes.

This dissertation contians studies in two related areas: Discrete Event Systems Theory and Queueing Network Theory. In the first line of study, deterministic discrete event systems are modeled by formal automata- like structures, and a hierarchy of morphic relations among them is developed. A canonical representation of stochastic discrete event systems in coordinate probability space is proposed, and a hierarchy of morphic relations among them is constructed by means of measure preserving transformations. In the second line of study, several operating characteristics of the class of Jackson queueing networks are investiaged. Included are: line sizes (network state), total service awarded to customers, and traffic processes on the arcs. Special emphasis is placed on rigorous derivations of results from solid mathematical and statistical foundations. In the process, a number of theoretical gaps in the extant theory of state equilibrium are closed, and Burke's Theorem is extended from M/M/1 queues to Jackson networks with single server nodes.

This dissertation contians studies in two related areas: Discrete Event Systems Theory and Queueing Network Theory. In the first line of study, deterministic discrete event systems are modeled by formal automata- like structures, and a hierarchy of morphic relations among them is developed. A canonical representation of stochastic discrete event systems in coordinate probability space is proposed, and a hierarchy of morphic relations among them is constructed by means of measure preserving transformations. In the second line of study, several operating characteristics of the class of Jackson queueing networks are investiaged. Included are: line sizes (network state), total service awarded to customers, and traffic processes on the arcs. Special emphasis is placed on rigorous derivations of results from solid mathematical and statistical foundations. In the process, a number of theoretical gaps in the extant theory of state equilibrium are closed, and Burke's Theorem is extended from M/M/1 queues to Jackson networks with single server nodes.

Thesis (MS)?Naval Postgraduate School, 1968

xx, 549 p. ; 24 cm

xx, 549 p. ; 24 cm

xx, 549 p. ; 24 cm

xx, 549 p. ; 24 cm

xx, 549 p. ; 24 cm

The problem of exact evaluation of the mean service cycle time in tandem systems of single-server queues with both infinite and finite buffers is considered. It is assumed that the interarrival and service times of customers form sequences of independent and identically distributed random variables with known mean values. We start with tandem queues with infinite buffers, and show that under the above assumptions, the mean cycle time exists. Furthermore, if the random variables which represent interarrival and service times have finite variance, the mean cycle time can be calculated as the maximum out from the mean values of these variables. Finally, obtained results are extended to evaluation of the mean cycle time in particular tandem systems with finite buffers and blocking.

The paper studies the problem of optimal adaptation of an M/M/1 queueing station, when the arrival rate lambda sub zero of customers shifts at unknown epoch, tau to a known value, lambda sub 1. The service intensity of the system starts at mu 0 and can be increased at most n times to mu 1 mu2 ... mu N. The cost structure consists of the cost of changing mu(i) to mu(j) (i+1 or = j or = N); of maintaining service at rate mu (per unit of time) and of holding customers at the station (per unit of time). Adaptation policies are constrained by the fact that mu can be only increased. A Bayes solution is derived, under the prior assumption that tau has an exponential distribution. This solution minimizes the total expected discounted cost for the entire future. (Author)

Bounds and comparisons for various performance measures in tandem queueing systems and loss systems are studied. Using a definition of variability of random variables, it is proven that for a tandem queueing system with n stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the variability of interarrival and service distributions decreases. A new sufficient condition for customer average delay to be smaller (larger) than time average delay in single server queues is also given.

Max-algebra models of tandem single-server queueing systems with both finite and infinite buffers are developed. The dynamics of each system is described by a linear vector state equation similar to those in the conventional linear systems theory, and it is determined by a transition matrix inherent in the system. The departure epochs of a customer from the queues are considered as state variables, whereas its service times are assumed to be system parameters. We show how transition matrices may be calculated from the service times, and present the matrices associated with particular models. We also give a representation of system performance measures including the system time and the waiting time of customers, associated with the models. As an application, both serial and parallel simulation procedures are presented, and their performance is outlined.

This paper presents a method for calculating steady state probabilities of $M|E_r|c|K$ queueing systems. The infinitesimal generator matrix is used to define all possible states in the system and their transition probabilities. While this matrix can be written down immediately for many other $M|PH|c|K$ queueing systems with phase-type service times (e.g. Coxian, Hypoexponential, \ldots), it requires a more careful analysis for systems with Erlangian service times. The constructed matrix may then be used to calculate steady state probabilities using an iterative algorithm. The resulting steady state probabilities can be used to calculate various performance measures, e.g. the average queue length. Additionally, computational issues of the implementation are discussed and an example from the field of telecommunication call-center queue length will be outlined to substantiate the applicability of these efforts. In the appendix, tables of the average queueing length given a specific number of service channels, traffic density, and system size are presented.

A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the system's state. We examine two versions of the problem: ``nonpreemptive,'' where service is uninterruptible, and ``preemptive,'' where service to a customer can be interrupted and then resumed, possibly at a different station. We study the problem in the asymptotic heavy traffic regime proposed by Halfin and Whitt, in which the arrival rates and the number of servers at each station grow without bound. The two versions of the problem are not, in general, asymptotically equivalent in this regime, with the preemptive version showing an asymptotic behavior that is, in a sense, much simpler. Under appropriate assumptions on the structure of the system we show: (i) The value function for the preemptive problem converges to $V$, the value of a related diffusion control problem. (ii) The two versions of the problem are asymptotically equivalent, and in particular nonpreemptive policies can be constructed that asymptotically achieve the value $V$. The construction of these policies is based on a Hamilton--Jacobi--Bellman equation associated with $V$.

A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the system's state. We examine two versions of the problem: ``nonpreemptive,'' where service is uninterruptible, and ``preemptive,'' where service to a customer can be interrupted and then resumed, possibly at a different station. We study the problem in the asymptotic heavy traffic regime proposed by Halfin and Whitt, in which the arrival rates and the number of servers at each station grow without bound. The two versions of the problem are not, in general, asymptotically equivalent in this regime, with the preemptive version showing an asymptotic behavior that is, in a sense, much simpler. Under appropriate assumptions on the structure of the system we show: (i) The value function for the preemptive problem converges to $V$, the value of a related diffusion control problem. (ii) The two versions of the problem are asymptotically equivalent, and in particular nonpreemptive policies can be constructed that asymptotically achieve the value $V$. The construction of these policies is based on a Hamilton--Jacobi--Bellman equation associated with $V$.

Four problem areas were studied. These are: (1) controlled heavy traffic queueing systems, (2) queueing systems with due dates, (3) backward-forward stochastic differential equations, and (4) Ginzburg-Landau equations and evolving interfaces. In areas (1) and (2), diffusion approximations were obtained for queues in heavy traffic. In (3), connections were established between quasi-linear partial differential equations and diffusion processes constructed via a new class of stochastic differential equations. Finally, (4) provides a study of the partial differential equation characterizing vortices in superconducting material in three dimensions.

Consider a production system composed of many processing stages (servers) through which the material (jobs) must pass in some prespecifled order. The performance of such a system is impaired by variations in servers behavior due either to failures or to fluctuations in service times. The effects of these variations can be mitigated by using intermediate storage spaces (buffers) between the servers. However, because of physical limitations on buffer spaces and variations in the service times, the flow of jobs through the system may get blocked. Queueing systems with blocking have a wider applicability in that they can also be used to model computer systems, telecommunication networks and distributed systems, to name a few applications.

Consider a production system composed of many processing stages (servers) through which the material (jobs) must pass in some prespecifled order. The performance of such a system is impaired by variations in servers behavior due either to failures or to fluctuations in service times. The effects of these variations can be mitigated by using intermediate storage spaces (buffers) between the servers. However, because of physical limitations on buffer spaces and variations in the service times, the flow of jobs through the system may get blocked. Queueing systems with blocking have a wider applicability in that they can also be used to model computer systems, telecommunication networks and distributed systems, to name a few applications.

In this paper we present and analyze a queueing-theoretical model for autonomous mobility-on-demand (MOD) systems where robotic, self-driving vehicles transport customers within an urban environment and rebalance themselves to ensure acceptable quality of service throughout the entire network. We cast an autonomous MOD system within a closed Jackson network model with passenger loss. It is shown that an optimal rebalancing algorithm minimizing the number of (autonomously) rebalancing vehicles and keeping vehicles availabilities balanced throughout the network can be found by solving a linear program. The theoretical insights are used to design a robust, real-time rebalancing algorithm, which is applied to a case study of New York City. The case study shows that the current taxi demand in Manhattan can be met with about 8,000 robotic vehicles (roughly 60% of the size of the current taxi fleet). Finally, we extend our queueing-theoretical setup to include congestion effects, and we study the impact of autonomously rebalancing vehicles on overall congestion. Collectively, this paper provides a rigorous approach to the problem of system-wide coordination of autonomously driving vehicles, and provides one of the first characterizations of the sustainability benefits of robotic transportation networks.

A model of a central processor (CPU) which services background applications in the presence of time critical activity is presented. The CPU is viewed as an M/M/1 queueing system subject to periodic interrupts by deterministic, time critical process. The Laplace transform of the distribution of service times for the background applications is developed. The use of state of the art queueing models for studying the background processing capability of time critical computer systems is discussed and the results of a model validation study which support this application of queueing models are presented.

It has been shown by A.-L. Barabasi that the priority based scheduling rules in single stage queuing systems (QS) generates fat tail behavior for the tasks waiting time distributions (WTD). Such fat tails are due to the waiting times of very low priority tasks which stay unserved almost forever as the task priority indices (PI) are "frozen in time" (i.e. a task priority is assigned once for all to each incoming task). Relaxing the "frozen in time" assumption, this paper studies the new dynamic behavior expected when the priority of each incoming tasks is time-dependent (i.e. "aging mechanisms" are allowed). For two class of models, namely 1) a population type model with an age structure and 2) a QS with deadlines assigned to the incoming tasks which is operated under the "earliest-deadline-first" policy, we are able to analytically extract some relevant characteristics of the the tasks waiting time distribution. As the aging mechanism ultimately assign high priority to any long waiting tasks, fat tails in the WTD cannot find their origin in the scheduling rule alone thus showing a fundamental difference between the present and the A.-L. Barabasi's class of models.

Satellites have an enormous potential for providing efficient communication links between many widely scattered ground stations. By interpreting the messages as customers in a queueing system, an unusual type of queueing model can be formulated to describe this process. The expected waiting time of the messages in the queue can then be derived for different configurations in order to guide the design of the system. (Author)

A regular Markov process is considered with continuous parameter, countable state space, and stationary transition probabilities, over which a class of traffic processes is defined. The feasibility that multiple traffic processes constitute mutually independent Poisson processes is investigated in some detail. A variety of independence conditions on the traffic process and the underlying Markov process are shown to be equivalent or sufficient to ensure Poisson related properties; these conditions include independent increments, renewal, weak pointwise independence, and pointwise independence. Two computational criteria for Poisson traffic are developed: a necessary condition in terms of weak pointwise independence, and a sufficient condition in terms of pointwise independence. The utility of these criteria is demonstrated by sample applications to queueing-theoretic models. It follows that, for the class of traffic processes as per this paper in a queueing-theoretic context, Kelly's notion of quasi-reversibility and Gelenbe and Muntz's notion of completeness are essentially equivalent to pointwise independence of traffic and state. The latter concept, however, is the most general one. The relevance of the theory developed to queueing network decomposition is also pointed out.