Heavy Traffic Limit Research Articles

Control of Polling in Presence of Vacations in Heavy Traffic with Applications to Satellite and Mobile Radio Systems

Consider a queueing system with many queues, each with its own input stream, but with only one server. The server must allocate its time among the queues to minimize or nearly minimize some cost criterion. The allocation of time among the queues is often called polling and is the subject of a large literature. Usually, it is assumed that the queues are always available, and the server can allocate at will. We consider the case where the queues are not always available due to disruption of the connection between them and the server. Such occurrences are common in wireless communications, where any of the mobile sources might become unavailable to the server from time to time due to obstacles, atmospheric or other effects. The possibility of such ``vacations' complicates the polling problem enormously. Due to the complexity of the basic problem we analyze it in the heavy traffic regime where the server has little idle time over the average requirements. It is shown that the suitable scaled total workloads converge to a controlled limit diffusion process with jumps. The jumps are due to the effects of the vacations. The control enters the dynamics only via its value just before a vacation begins; hence it is only via the jump value that the control affects the dynamics. This type of model has not received much attention. The individual queued workloads and job numbers can be recovered (asymptotically) from the limit scaled workload. This state space collapse is critical for the effective numerical and analytical work, since the limit process is one dimensional. It is also shown, under appropriate conditions, that the arrival process during a vacation can be approximated by the scaled "fluid" process. With a suitable nonlinear discounted cost rate, it is shown that the optimal costs for the physical problems converge to that for the limit problem as the traffic intensity approaches its heavy traffic limit. Explicit solutions are obtained in some simple but important cases, and the $c\mu$-rule is asymptotically optimal if there are no vacations. The stability of the queues is analyzed via a perturbed Liapunov function method, under quite general conditions on the data. Finally, we extend the results to unreliable channels where the data might be received with errors and need to be retransmitted.

A functional central limit theorem for Markov additive processes with an application to the closed Lu–Kumar network

A functional central limit theorem for a class of time-homogeneous continuous-time Markov processes (X,Y) is proved. The process X is a positive recurrent Markov process on a countable-state space and the process Y has conditionally independent increments given X. The pair (X,Y) is called a Markov additive process. This paper unifies and generalizes several functional central limit theorems for Markov additive processes. An explicit expression for the variance parameter of the limit process is calculated using the local characteristics of the X process. The functional central limit theorem is then used to prove a heavy traffic limit theorem for the closed Lu–Kumar network.

Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy

This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to infinite capacity buffers and two (nonidentical) servers working in parallel. One server can only process jobs from one buffer. The service time distribution may depend on the buffer being served and the server providing the service. The system manager dynamically schedules waiting jobs onto available servers. We consider a parameter regime in which the system satisfies both a heavy traffic condition and a resource pooling condition. Our cost function is a mean cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. We first review the analytic solution of the Brownian control problem (formal heavy traffic approximation) for this system. We "interpret" this solution by proposing a threshold control policy for use in the original parallel server system. We show that this policy is asymptotically optimal in the heavy traffic limit and the limiting cost is the same as the optimal cost in the Brownian control problem. The techniques developed here are expected to be useful for analyzing the performance of threshold-type policies in more complex multiserver systems.

Parallel Scheduling of Multiclass M/M/m Queues: Approximate and Heavy-Traffic Optimization of Achievable Performance

We address the problem of scheduling a multiclass M/M/m queue with Bernoulli feedback on m parallel servers to minimize time-average linear holding costs. We analyze the performance of a heuristic priority-index rule, which extends Klimov's optimal solution to the single-server case: servers select preemptively customers with larger Klimov indices. We present closed-form suboptimality bounds (approximate optimality) for Klimov's rule, which imply that its suboptimality gap is uniformly bounded above with respect to (i) external arrival rates, as long as they stay within system capacity; and (ii) the number of servers. It follows that its relative suboptimality gap vanishes in a heavy-traffic limit, as external arrival rates approach system capacity (heavy-traffic optimality). We obtain simpler expressions for the special no-feedback case, where the heuristic reduces to the classical cμ rule. Our analysis is based on comparing the expected cost of Klimov's rule to the value of a strong linear programming (LP) relaxation of the system's region of achievable performance of mean queue lengths. In order to obtain this relaxation, we derive and exploit a new set of work decomposition laws for the parallel-server system. We further report on the results of a computational study on the quality of the cμ rule for parallel scheduling.

The Reflection Map with Discontinuities

We study the multidimensional reflection map on the spaces D([0, T], ℝk) and D([0, ∞), ℝk) of right-continuous ℝk-valued functions on [0, T] or [0, ∞) with left limits, endowed with variants of the Skorohod (1956) M1 topology. The reflection map was used with the continuous mapping theorem by Harrison and Reiman (1981) and Reiman (1984) to establish heavy-traffic limit theorems with reflected Brownian motion limit processes for vector-valued queue length, waiting time, and workload stochastic processes in single-class open queueing networks. Since Brownian motion and reflected Brownian motion have continuous sample paths, the topology of uniform convergence over bounded intervals could be used for those results. Variants of the M1 topologies are needed to obtain alternative discontinuous limits approached gradually by the converging processes, as occurs in stochastic fluid networks with bursty exogenous input processes, e.g., with on-off sources having heavy-tailed on periods or off periods (having infinite variance). We show that the reflection map is continuous at limits without simultaneous jumps of opposite sign in the coordinate functions, provided that the product M1 topology is used. As a consequence, the reflection map is continuous with the product M1 topology at all functions that have discontinuities in only one coordinate at a time. That continuity property also holds for more general reflection maps and is sufficient to support limit theorems for stochastic processes in most applications. We apply the continuity of the reflection map to obtain limits for buffer-content stochastic processes in stochastic fluid networks.

Heavy Traffic Analysis of Dynamic Cyclic Policies: A Unified Treatment of the Single Machine Scheduling Problem

This paper examines how setups, due dates, and the mix of standardized and customized products affect the scheduling of a single machine operating in a dynamic and stochastic environment. We restrict ourselves to the class of dynamic cyclic policies, where the machine busy/idle policy and lot-sizing decisions are controlled in a dynamic fashion, but different products must be produced in a fixed sequence. As in earlier work, we conjecture that an averaging principle holds for this queueing system in the heavy traffic limit, and optimize over the class of dynamic cyclic policies. The results allow for a detailed discussion of the interactions between the due-date, setup, and product mix facets of the problem.

Heavy Traffic Limits for Some Queueing Networks

Using a slight modification of the framework of Bramson [7] and Williams [54], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single-station systems operating under first-in-first-out (FIFO), generalized-head-of-the-line proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines. The next two families are reentrant lines that operate under first-buffer-first-serve (FBFS) and last-buffer-first-serve (LBFS) service disciplines; the last family consists of certain two-station, five-class networks operating under an SBP service discipline. Some of these heavy traffic limits have appeared earlier in the literature; our new proofs demonstrate the significant simplifications that can be achieved in the present setting.

A Multiclass Queue in Heavy Traffic with Throughput Time Constraints: Asymptotically Optimal Dynamic Controls

Consider a single-server queueing system with K job classes, each having its own renewal input process and its own general service time distribution. Further suppose the queue is in heavy traffic, meaning that its traffic intensity parameter is near the critical value of one. A system manager must decide whether or not to accept new jobs as they arrive, and also the order in which to serve jobs that are accepted. The goal is to minimize penalties associated with rejected jobs, subject to upper bound constraints on the throughput times for accepted jobs; both the penalty for rejecting a job and the bound on the throughput time may depend on job class. This problem formulation does not make sense in a conventional queueing model, because throughput times are random variables, but we show that the formulation is meaningful in an asymptotic sense, as one approaches the heavy traffic limit under diffusion scaling. Moreover, using a method developed recently by Bramson and Williams, we prove that a relatively simple dynamic control policy is asymptotically optimal in this framework. Our proposed policy rejects jobs from one particular class when the server's nominal workload is above a threshold value, accepting all other arrivals; and the sequencing rule for accepted jobs is one that maintains near equality of the relative backlogs for different classes, defined in a natural sense.

Asymptotic behaviour of the tandem queueing system with identical service times at both queues

Consider a tandem queue consisting of two single-server queues in series, with a Poisson arrival process at the first queue and arbitrarily distributed service times, which for any customer are identical in both queues. For this tandem queue, we relate the tail behaviour of the sojourn time distribution and the workload distribution at the second queue to that of the (residual) service time distribution. As a by-result, we prove that both the sojourn time distribution and the workload distribution at the second queue are regularly varying at infinity of index 1−ν, if the service time distribution is regularly varying at infinity of index −ν (ν>1). Furthermore, in the latter case we derive a heavy-traffic limit theorem for the sojourn time S (2) at the second queue when the traffic load ρ↑ 1. It states that, for a particular contraction factor Δ (ρ), the contracted sojourn time Δ (ρ) S (2) converges in distribution to the limit distribution H(·) as ρ↑ 1 where .

Diffusion Approximations for Some Multiclass Queueing Networks with FIFO Service Disciplines

The diffusion approximation is proved for a class of multiclass queueing networks under FIFO service disciplines. In addition to the usual assumptions for a heavy traffic limit theorem, a key condition that characterizes this class is that a J × J matrix G, known as the workload contents matrix, has a spectral radius less than unity, where J represents the number of service stations. The (j, 𝓁)th component of matrix Gcan be interpreted as the long-run average amount of future work for station j that is embodied in a unit of immediate work at station 𝓁. This class includes, as a special case, the feedforward multiclass queueing network and the Rybko-Stolyar network under FIFO service discipline.A new approach is taken in establishing the diffusion limit theorem. The traditional approach is based on an oblique reflection mapping, but such a mapping is not well defined for the network under consideration. Our approach takes two steps: Arst establishing the C-tightness of the scaled queueing processes, and then completing the proof for the convergence of the scaled queueing processes by invoking the weak uniqueness for the limiting processes, which are semimartingale reflecting Brownian motions.

Strong approximations for multiclass feedforward queueing networks

This paper derives the strong approximation for a multiclass queueing network,where jobs after service completion can only move to a downstream service station. Job classes are partitioned into groups. Within a group, jobs are served in the order of arrival; that is, a first-in first-out (FIFO) discipline is in force, and among groups, jobs are served under a preassigned preemptive priority discipline. We obtain the strong approximation for the network through an inductive application of an input–output analysis for a single-station queue. Specifically, we show that if the input data (i.e., the arrival and the service processes) satisfy an approximation (such as the functional law-of-iterated logarithm approximation or the strong approximation), then the output data (i.e., the departure processes) and the performance measures (such as the queue length, the workload and the sojourn time processes) satisfy a similar approximation. Based on the strong approximation, some procedures are proposed to approximate the stationary distribution of various performance measures of the queueing network. Our work extends and complements the existing work of Peterson and Harrison and Williams on the feedforward queueing network. The numeric examples show that strong approximation provides a better approximation than that suggested by a straightforward interpretation of the heavy traffic limit theorem.

A join the shorter queue model in heavy traffic

We prove a new heavy traffic limit result for a simple queueing network under a ‘join the shorter queue’ policy, with the amount of traffic which has a routeing choice tending to zero as heavy traffic is approached. In this limit, the system considered does not exhibit state space collapse as in previous work by Foschini and Salz, and Reiman, but there is nevertheless some resource pooling gain over a policy of random routeing.

A join the shorter queue model in heavy traffic

We prove a new heavy traffic limit result for a simple queueing network under a ‘join the shorter queue’ policy, with the amount of traffic which has a routeing choice tending to zero as heavy traffic is approached. In this limit, the system considered does not exhibit state space collapse as in previous work by Foschini and Salz, and Reiman, but there is nevertheless some resource pooling gain over a policy of random routeing.

The Stochastic Economic Lot Scheduling Problem: Heavy Traffic Analysis of Dynamic Cyclic Policies

We consider two queueing control problems that are stochastic versions of the economic lot scheduling problem: A single server processes N customer classes, and completed units enter a finished goods inventory that services exogenous customer demand. Unsatisfied demand is backordered, and each class has its own general service time distribution, renewal demand process, and holding and backordering cost rates. In the first problem, a setup cost is incurred when the server switches class, and the setup cost is replaced by a setup time in the second problem. In both problems we employ a long-run average cost criterion and restrict ourselves to a class of dynamic cyclic policies, where idle periods and lot sizes are state-dependent, but the N classes must be served in a fixed sequence. Motivated by existing heavy traffic limit theorems, we make a time scale decomposition assumption that allows us to approximate these scheduling problems by diffusion control problems. Our analysis of the approximating setup cost problem yields a closed-form dynamic lot-sizing policy and a computational procedure for an idling threshold. We derive structural results and an algorithmic procedure for the setup time problem. A computational study compares the proposed policy and several alternative policies to the numerically computed optimal policy.

SECOND MOMENTS OF THE WAITING TIME IN SYMMETRIC POLLING SYSTEMS

Multiple queue, cyclic service systems (called polling systems) have often been used as performance evaluation models in communication and production systems with cyclic resource allocation. However, most research has focused only on the mean waiting times due to prohibitively growing complexity in computing higher-order moments of the waiting time. This paper presents the explicit expressions for the second moments of the waiting time in symmetric exhaustive and gated service systems of two, three, and four queues with Poisson arrival processes. Numerical comparison reveals that they are ordered in the number of queues (increasingly/decreasingly for exhaustive/gated service systems, respectively) and bounded fairly tightly by those for single queue systems and by those for systems with infinitely many queues. Conjecture of their heavy traffic limits is also made.

Sojourn times in a processor sharing queue with service interruptions

We study the sojourn times of customers in an M/M/1 queue with the processor sharing service discipline and a server that is subject to breakdowns. The lengths of the breakdowns have a general distribution, whereas the “on-periods” are exponentially distributed. A branching process approach leads to a decomposition of the sojourn time, in which the components are independent of each other and can be investigated separately. We derive the Laplace–Stieltjes transform of the sojourn-time distribution in steady state, and show that the expected sojourn time is not proportional to the service requirement. In the heavy-traffic limit, the sojourn time conditioned on the service requirement and scaled by the traffic load is shown to be exponentially distributed. The results can be used for the performance analysis of elastic traffic in communication networks, in particular, the ABR service class in ATM networks, and best-effort services in IP networks.

Heavy traffic limits associated with M/G/∞ input processes

We study the heavy traffic regime of a discrete-time queue driven by correlated inputs, namely the M/G/∞ input processes of Cox. We distinguish between M/G/∞ processes with short- and long-range dependence, identifying in each case the appropriate heavy traffic scaling that results in a nondegenerate limit. As expected, the limits we obtain for short-range dependent inputs involve the standard Brownian motion. Of particular interest are the conclusions for the long-range dependent case: the normalized queue length can be expressed as a function not of a fractional Brownian motion, but of an a-stable, 1/a self-similar independent increment Levy process. The resulting buffer content distribution in heavy traffic is expressed through a Mittag–Leffler special function and displays a hyperbolic decay of power 1-a. Thus, M/G/∞ processes already demonstrate that under long-range dependence, fractional Brownian motion does not necessarily assume the ubiquitous role that standard Brownian motion plays in the short-range dependence setup.

Optimal control of assignment of jobs to processors under heavy traffic

The paper is concerned with the optimal control of the assignment of jobs from several arriving random streams to one of a bank of processors. Owing to the difficulty of the general problem, a heavy traffic approach is used. The required work depends on the processor to which it is assigned. The information that the assignment can be based on is quite flexible, and several information structures (data on which the control is based) are considered. The assignment can be made on arrival or when the job is to be processed. There can be bursty arrivals (the bursts depending on randomly varying environmental factors), rather general nonlinear cost functions and other complications. It is shown, under reasonably general conditions, that the optimal costs for the physical systems converge to the optimal cost for the heavy traffic limit problem, as the heavy traffic parameter goes to its limit. Numerical data is presented to illustrate some of the potential uses of the limit process for obtaining optimal contros, or controls satisfying optimal tradeoffs among competing criteria. The methods of proof are quite powerful tools for such optimal control problems

Heavy Traffic Analysis of the Dynamic Stochastic Inventory-Routing Problem

We analyze three queueing control problems that model a dynamic stochastic distribution system, where a single capacitated vehicle serves a finite number of retailers in a make-to-stock fashion. The objective in each of these vehicle routing and inventory problems is to minimize the long run average inventory (holding and backordering) and transportation cost. In all three problems, the controller dynamically specifies whether a vehicle at the warehouse should idle or embark with a full load. In the first problem, the vehicle must travel along a prespecified (TSP) tour of all retailers, and the controller dynamically decides how many units to deliver to each retailer. In the second problem, the vehicle delivers an entire load to one retailer (direct shipping) and the controller decides which retailer to visit next. The third problem allows the additional dynamic choice between the TSP and direct shipping options. Motivated by existing heavy traffic limit theorems, we make a time scale decomposition assumption that allows us to approximate these queueing control problems by diffusion control problems, which are explicitly solved in the fixed route problems, and numerically solved in the dynamic routing case. Simulation experiments confirm that the heavy traffic approximations are quite accurate over a broad range of problem parameters. Our results lead to some new observations about the behavior of this complex system.

A new heavy traffic limit for the asymmetric shortest queue problem

We consider the classic shortest queue problem in the heavy traffic limit. We assume that the second server works slowly and that the service rate of the first server is nearly equal to the arrival rate. Solving for the (asymptotic) joint steady state queue length distribution involves analyzing a backward parabolic partial differential equation, together with appropriate side conditions. We explicitly solve this problem. We thus obtain a two-dimensional approximation for the steady state queue length probabilities.