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.     

Queueing Networks of Random Link Topology: Stationary Dynamics of Maximal Throughput Schedules

In this paper, we study the stationary dynamics of a processing system comprised of several parallel queues and a single server of constant rate. The connectivity of the server to each queue is randomly modulated, taking values 1 (connected) or 0 (severed). At any given time, only the currently connected queues may receive service. A key issue is how to schedule the server on the connected queues in order to maximize the system throughput. We investigate two dynamic schedules, which are shown to stabilize the system under the highest possible traffic load, by scheduling the server on the connected queue of maximum backlog (workload or job number). They are analyzed under stationary ergodic traffic flows and connectivity modulation. The results also extend to the more general case of random server rate.We then investigate the dynamics of acyclic (feed-forward) queueing networks with nodes of the previous type. Their links (connectivities) are stochastically modulated, inducing fluctuating network topologies. We focus on the issue of network throughput and show that it is maximized by simple node server schedules. Rate ergodicity of the traffic flows traversing the network is established, allowing the computation of the maximal throughput.Queueing networks of random topology model several practical systems with unreliable service, including wireless communication networks with extraneous interference, flexible manufacturing systems with failing components, production management under random availability of resources etc.Research supported in part by the National Science Foundation.This revised version was published online in June 2005 with corrected coverdate     

Dynamic scheduling in multiclass queueing networks: Stability under discrete-review policies

This paper describes a family of discrete-review policies for scheduling open multiclass queueing networks. Each of the policies in the family is derived from what we call a dynamic reward function: such a function associates with each queue length vector q and each job class k a positive value r _k (q), which is treated as a reward rate for time devoted to processing class k jobs. Assuming that each station has a traffic intensity parameter less than one, all policies in the family considered are shown to be stable. In such a policy, system status is reviewed at discrete points in time, and at each such point the controller formulates a processing plan for the next review period, based on the queue length vector observed. Stability is proved by combining elementary large deviations theory with an analysis of an associated fluid control problem. These results are extended to systems with class dependent setup times as well as systems with alternate routing and admission control capabilities. This revised version was published online in June 2006 with corrections to the Cover Date.     

Symmetry property of the throughput in closed tandem queueing networks with finite buffers

In this paper we consider closed tandem queueing networks with finite buffers and blocking before service. With this type of blocking, a server is allowed to start processing a job only if there is an empty space in the next buffer. It was recently conjectured that the throughput of such networks is symmetrical with respect to the population of the network. That is, the throughput of the network with population N is the same as that with population C − N, where C is the total number of buffer spaces in the network. The main purpose of this paper is to prove this result in the case where the service time distributions are of phase type (PH-distribution). The proof is based on the comparison of the sample paths of the network with populations N and C − N. Finally, we also show that this symmetry property is related to a reversibility property of this class of networks.     

On stationary tandem queueing networks with job feedback

The class of tandem queueing networks with job feedback is studied under stationarity conditions on the arrival and service times sequences. Each job, after completing service in the last queue, is fed back (rerouted) to the first one, a random number of times, before leaving the system. The average execution time per job is exactly computed, as the number of jobs becomes large, and is minimized under mild conditions. The degree of parallelism achieved in the processing is also computed. The issue of rate-stability of the system is then considered. The network is defined to be rate-stable iff the job departure rate is equal to the job arrival rate; that depends heavily on the dynamic feedback policy we employ to place rerouted jobs in specific places of the front queue buffer of the network. The condition under which the network is rate-stable is specified, and a dynamic feedback policy is constructed, which rate-stabilizes the system under the maximum possible job arrival rate; thus, it maximizes the dynamic throughput of the network. Other related results concerning the performance of tandem networks with feedback are obtained.Research supported in part by grants NSF-DDM-RIA-9010778, NSF-NCR-9116268, NSF-NCR-NYI-9258 507, by an AT&T Foundation grant and a GTE Fellowship.     

抢占型优先服务机制下多类排队网络的扩散逼近

证明一个满负荷交通极限定理以证实在抢占型优先服务机制下多类排队网络的扩散逼近,进而为该系统提供有效的随机动力学模型．所研究的排队网络典型地出现在现代通讯系统中高速集成服务分组数据网络,其中包含分组数据包的若干交通类型,每个类型涉及若干工作处理类（步骤）,并且属于同一交通类型的工作在可能接受服务的每一个网站被赋予相同的优先权等级,更进一步地,在整个网络中,属于不同交通类型的分组数据包之间无交互路由．     

Linear preselective policies for stochastic project scheduling

In the context of stochastic resource-constrained project scheduling we introduce a novel class of scheduling policies, the linear preselective policies. They combine the benefits of preselective policies and priority policies; two classes that are well known from both deterministic and stochastic scheduling. We study several properties of this new class of policies which indicate its usefulness for computational purposes. Based on a new representation of preselective policies as and/or precedence constraints we derive efficient algorithms for computing earliest job start times and state a necessary and sufficient dominance criterion for preselective policies.  A computational experiment based on 480 instances empirically validates the theoretical findings.     

A two-server queueing system with periodic and credit-based server availabilities

A discrete-time, two-server queueing system is studied in this paper. The service time of a customer (cell) is fixed and equal to one time unit. Server 1 provides for periodic service of the queue (periodT). Server 2 provides for service only when server 1 is unavailable and provided that the associated service credit is nonzero. The resulting system is shown to model the queueing behavior of a network user which is subject to traffic regulation for congestion avoidance in high speed ATM networks. A general methodology is developed for the study of this queueing system, based on renewal theory. The dimensionality of the developed model is independent ofT;T increases with the network speed. The cell loss probabilities are computed in the case of finite capacity queue.Research supported by the National Science Foundation under grant NCR-9011962.     

Heavy traffic resource pooling in parallel‐server systems

We consider a queueing system with r non‐identical servers working in parallel, exogenous arrivals into m different job classes, and linear holding costs for each class. Each arrival requires a single service, which may be provided by any of several different servers in our general formulation; the service time distribution depends on both the job class being processed and the server selected. The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs onto available servers. A linear program involving only first‐moment data (average arrival rates and mean service times) is used to define heavy traffic for a system of this form, and also to articulate a condition of overlapping server capabilities which leads to resource pooling in the heavy traffic limit. Assuming that the latter condition holds, we rescale time and state space in standard fashion, then identify a Brownian control problem that is the formal heavy traffic limit of our rescaled scheduling problem. Because of the assumed overlap in server capabilities, the limiting Brownian control problem is effectively one‐dimensional, and it admits a pathwise optimal solution. That is, in the limiting Brownian control problem the multiple servers of our original model merge to form a single pool of service capacity, and there exists a dynamic control policy which minimizes cumulative cost incurred up to any time t with probability one. Interpreted in our original problem context, the Brownian solution suggests the following: virtually all backlogged work should be held in one particular job class, and all servers can and should be productively employed except when the total backlog is small. It is conjectured that such ideal system behavior can be approached using a family of relatively simple scheduling policies related to the cμ rule. This revised version was published online in June 2006 with corrections to the Cover Date.     

Throughput maximization for two station tandem systems: a proof of the Andradóttir–Ayhan conjecture

We study a tandem queueing network with two stations, M heterogeneous flexible servers, and a finite intermediate buffer. The objective is to dynamically assign the servers to the stations in order to maximize the throughput of the system. The form of the optimal policy for M≤3 was derived in two previous papers. In one of those papers, Andradóttir and Ayhan (Operations Research 53:516–531, 2005) provide a conjecture on the form of the optimal policy for M≥4. We prove their conjecture in this paper, showing that the optimal policy is defined by monotone thresholds and the ratios of the service rates among the servers. For M>1, we also prove that the optimal policy always uses the entire intermediate buffer.     

Continuous-Review Tracking Policies for Dynamic Control of Stochastic Networks

This paper is concerned with dynamic control of stochastic processing networks. Specifically, it follows the so called heavy traffic approach, where a Brownian approximating model is formulated, an associated Brownian optimal control problem is solved, the solution of which is then used to define an implementable policy for the original system. A major challenge is the step of policy translation from the Brownian to the discrete network. This paper addresses this problem by defining a general and easily implementable family of continuous-review tracking policies. Each such policy has the following structure: at each point in time t, the controller observes the current vector of queue lengths q and chooses (i) a target position z(q) of where the system should be at some point in the near future, say at time t+l, and (ii) an allocation vector v(q) that describes how to split the server's processing capacity amongst job classes in order to steer the state from q to z(q). Implementation of such policies involves the enforcement of small safety stocks. In the context of the heavy traffic approach, the solution of the approximating Brownian control problem is used in selecting the target state z(q). The proposed tracking policy is shown to be asymptotically optimal in the heavy traffic limiting regime, where the Brownian model approximation becomes valid, for multiclass queueing networks that admit orthant Brownian optimal controls; this is a form of pathwise, or greedy, optimality. Several extensions are discussed.     

Bulk input queues with hysteretic control

This paper deals with a bulk input-batch service queueing system and (r,N)-hysteretic control, i.e., under N-policy and r-quorum discipline. The model also includes state dependent service. The goal of such global control is to reduce the waste of server capacity (r-quorum), an unwanted number of switchovers between idle and busy modes (N-policy), and the queue length (by means of variable service rates). The analysis of the system is based on first excess level technique developed by the second author. This approach enables the authors to obtain major characteristics for the queueing process in a closed analytical form. This revised version was published online in June 2006 with corrections to the Cover Date.     

Counterexamples for comparisons of queues with finite waiting rooms

We constructG/G/1/k queueing models that fail to have anticipated monotonicity properties with respect to the capacityk. In one model the long-run average number of customers in the system is arbitrarily close to the capacityk, but it decreases to an arbitrarily small value when the capacity is increased. In another model the throughput is arbitrarily close to the arrival rate when the capacity isk, but the throughput decreases to an arbitrarily small value when the capacity is increased. These examples involving non i.i.d. service times, which are associated with external arrivals instead of being assigned when service begins, show that stochastic assumptions and arguments involving more than direct sample-path comparisons are essential for obtaining useful bounds and positive comparison results.     

A duality approach to admission and scheduling controls of queues

This paper studies theadmission andscheduling control problem in anM/M/2 queueing system with nonidentical processors. Admission control renders when a newly arrived job should be accepted, whereas scheduling control determines when an available processor should be utilized. The system received a rewardR when a job completes its service and pays a unit holding costC while a job is in the system. The main goal of the paper is to obtain the admission/scheduling policy that maximizes the expected discounted and long-run average profits (reward minus cost). We convert the system into its dual, a stochastically identical system subject toexpulsion/scheduling control, and prove that the individually optimal policy in the dual system is socially optimal in the original system. In contrast with the dynamic programming (DP) technique which considers the system as a whole, we adopt the viewpoint of an individual job and analyze the impact of its behavior on the social outcome. The key properties which simplify the analysis are that under the individually optimal policy the profit of a job under the preemptive last-come first-priority service discipline (LCFP-P) is independent of jobs arrived earlier than itself and that the system is insensitive to service discipline imposed. The former makes possible to bypass complex dynamic programming analyses and the latter serves as a vehicle in connecting the social and individual optimality. We also exploit system operational characteristics under LCFP-P to obtain simple and close approximations of the optimal thresholds.     

Effects of service disciplines inG/GI/s queueing systems

Transient extremal properties of some service disciplines are established in theG/GI/s queueing system for the minimization and maximization of the expectations of the Schur convex functions, convex symmetric functions and the sums of convex functions of the waiting times, response times, lag times and latenesses. When resequencing is required in the system, the FCFS and LCFS disciplines are shown to minimize and maximize, respectively, the expectations of any increasing functions of the end-to-end delays. All of these results are presented in terms of stochastic orderings. The paper concludes with extensions of the results to the stationary regime and to tandem as well as general queueing networks.This work was supported in part by the National Science Foundation under grant ASC 88-8802764.The work of this author was also partially supported by CEC DG-XIII under the ESPRIT-BRA grant QMIPS.     

Analysis of job assignment with batch arrivals among heterogeneous servers

We revisit the problem of job assignment to multiple heterogeneous servers in parallel. The system under consideration, however, has a few unique features. Specifically, repair jobs arrive to the queueing system in batches according to a Poisson process. In addition, servers are heterogeneous and the service time distributions of the individual servers are general. The objective is to optimally assign each job within a batch arrival to minimize the long-run average number of jobs in the entire system. We focus on the class of static assignment policies where jobs are routed to servers upon arrival according to pre-determined probabilities. We solve the model analytically and derive the structural properties of the optimal static assignment. We show that when the traffic is below a certain threshold, it is better to not assign any jobs to slower servers. As traffic increases (either due to an increase in job arrival rate or batch size), more slower servers will be utilized. We give an explicit formula for computing the threshold. Finally we compare and evaluate the performance of the static assignment policy to two dynamic policies, specifically the shortest expected completion policy and the shortest queue policy.     

Dynamic scheduling for switched processing systems with substantial service-mode switching times

Switched Processing Systems (SPS) represent canonical models for many communication and computer systems. Over the years, much research has been devoted to developing the best scheduling policies to optimize the various performance metrics of interest. These policies have mostly originated from the well-known MaxWeight discipline, which at any point in time switches the system into the service mode possessing “maximal matching” with the system state (e.g., queue-length, workload, etc.). However, for simplicity it is often assumed that the switching times between service modes are “negligible”—but this proves to be impractical in some applications. In this study, we propose a new scheduling strategy (called the Dynamic Cone Policy) for SPS, which includes substantial service-mode switching times. The goal is to maximize throughput and maintain system stability under fairly mild stochastic assumptions. For practical purposes, an extended scheduling strategy (called the Practical Dynamic Cone Policy) is developed to reduce the computational complexity of the Dynamic Cone Policy and at the same time mitigate job delay. A simulation study shows that the proposed practical policy clearly outperforms another throughput-maximizing policy called BatchAdapt, both in terms of the average and the 95th percentile of job delay for various types of input traffic.     

Probability analysis of the repairable queueing systemM/G(E k /H)/1

The repairable queueing system (RQS) in which the server has an exponential lifetime distribution has been studied in several articles [1–4]. Here, we deal with the new RQSM/G(E _k /H)/1 in which the lifetime distribution of the server is Erlangian. By forming a vector Markov process, i.e. by using the method of supplementary variables, we obtained some system characters, the reliability indices of the server, and the time distribution of a customer spent on the server. For this RQS, the generalized service time distribution of each customer will depend on the remainder life of the server. Based on this, a new kind of queues, for which the service time distributions are chosen by the customers in some stochastic manner, appears in queueing theory.Project supported by the National Natural Science Foundation of China.     

A matrix-analytic solution for the DBMAP/PH/1 priority queue

Priority queueing models have been commonly used in telecommunication systems. The development of analytically tractable models to determine their performance is vitally important. The discrete time batch Markovian arrival process (DBMAP) has been widely used to model the source behavior of data traffic, while phase-type (PH) distribution has been extensively applied to model the service time. This paper focuses on the computation of the DBMAP/PH/1 queueing system with priorities, in which the arrival process is considered to be a DBMAP with two priority levels and the service time obeys a discrete PH distribution. Such a queueing model has potential in performance evaluation of computer networks such as video transmission over wireless networks and priority scheduling in ATM or TDMA networks. Based on matrix-analytic methods, we develop computation algorithms for obtaining the stationary distribution of the system numbers and further deriving the key performance indices of the DBMAP/PH/1 priority queue. AMS subject classifications: 60K25 · 90B22 · 68M20 The work was supported in part by grants from RGC under the contracts HKUST6104/04E, HKUST6275/04E and HKUST6165/05E, a grant from NSFC/RGC under the contract N_HKUST605/02, a grant from NSF China under the contract 60429202.