Round robin scheduling of heterogeneous parallel servers in heavy traffic

We consider scheduling for heterogeneous server systems, where tasks arrive according to a Poisson process, with their processing requirements following a discrete distribution with finite support. For a system with a dispatcher and several heterogeneous servers, we propose an optimized multi-layered round robin routing policy followed by shortest remaining processing time scheduling at each server. Using a heavy traffic approximation, we show that the proposed policy performs as well as the optimal scheduling policy for a heterogeneous servers system with a single queue (no routing) in heavy traffic. Additional simulation results suggest that such policies will be effective in more general settings.     

Multi-layered round robin routing for parallel servers

We study a system of several identical servers in parallel, where a routing decision must be made immediately on a job’s arrival. Jobs arrive according to a Poisson process, with their processing times following a discrete distribution with finite support. The processing time of a job is known on arrival and may be used in the routing decision. We propose a policy consisting of multi-layered round robin routing followed by shortest remaining processing time scheduling at the servers. This policy is shown to have a heavy traffic limit that is identical to one in which there is a single queue (no routing) and optimal heavy traffic scheduling. In light traffic, we show that the performance of this policy is worse than round robin routing followed by shortest remaining processing time scheduling. We also quantify the difference between round robin and multi-layered round robin routing, which in turn yields insights on the relative importance of routing versus (local) scheduling in such systems. AMS subject classifications: 68M20 · 60K25 (Work done while both authors were visitors at EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.)     

Asymptotic optimality of the Round–Robin policy in multipath routing with resequencing

We consider a model of a multipath routing system where arriving customers are routed to a set of identical, parallel, single server queues according to balancing policies operating without state information. After completion of service, customers are required to leave the system in their order of arrival, thus incurring an additional resequencing delay. We are interested in minimizing the end-to-end delay (including time at the resequencing buffer) experienced by arriving customers. To that end we establish the optimality of the Round–Robin routing assignment in two asymptotic regimes, namely heavy and light traffic: In heavy traffic, the Round–Robin customer assignment is shown to achieve the smallest (in the increasing convex stochastic ordering) end-to-end delay amongst all routing policies operating without queue state information. In light traffic, and for the special case of Poisson arrivals, we show that Round–Robin is again an optimal (in the strong stochastic ordering) routing policy. We illustrate the stochastic comparison results by several simulation examples. The work of the first author was supported through an ARCHIMEDES grant by the Greek Ministry of Education. The work of the second author was prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U.S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government.     

A stochastic and dynamic model for the single-vehicle pick-up and delivery problem

In this paper a stochastic and dynamic model for the Pick-up and Delivery Problem is developed and analyzed. Demands for service arrive according to a Poisson process in time. The pick-up locations of the demands are independent and uniformly distributed over a service region. A single vehicle must transport the demands from the pick-up to the delivery location. Once a demand has been picked up it can only be dropped off at its desired delivery location. The delivery locations are independent and uniformly distributed over the region, and they are independent of the pick-up locations. The objective is to minimize the expected time in the system for the demands. Unit-capacity vehicle and multiple-capacity vehicle variations are considered. For each variation, bounds on the performance of the routing policies are derived for light and heavy traffic. The policies are analyzed using both analytical methods and simulation.     

Decomposition results for general polling systems and their applications

In this paper we derive decomposition results for the number of customers in polling systems under arbitrary (dynamic) polling order and service policies. Furthermore, we obtain sharper decomposition results for both the number of customers in the system and the waiting times under static polling policies. Our analysis, which is based on distributional laws, relaxes the Poisson assumption that characterizes the polling systems literature. In particular, we obtain exact decomposition results for systems with either Mixed Generalized Erlang (MGE) arrival processes, or asymptotically exact decomposition results for systems with general renewal arrival processes under heavy traffic conditions. The derived decomposition results can be used to obtain the performance analysis of specific systems. As an example, we evaluate the performance of gated Markovian polling systems operating under heavy traffic conditions. We also provide numerical evidence that our heavy traffic analysis is very accurate even for moderate traffic. This revised version was published online in June 2006 with corrections to the Cover Date.     

Towards minimum loss job routing to parallel heterogeneous multiserver queues via index policies

This paper considers a Markovian model for the optimal dynamic routing of homogeneous traffic to parallel heterogeneous queues, each having its own finite input buffer and server pool, where buffer and server-pool sizes, as well as service rates, may differ across queues. The main goal is to identify a heuristic index-based routing policy with low complexity that consistently attains a nearly minimum average loss rate (or, equivalently, maximum throughput rate). A second goal is to compare alternative policies, with respect to computational demands and empirical performance. A novel routing policy that can be efficiently computed is developed based on a second-order extension to Whittle’s restless bandit (RB) index, since the latter is constant for this model. New results are also given for the more computationally demanding index policy obtained via policy improvement (PI), including that it reduces to shortest queue routing under symmetric buffer and server-pool sizes. A numerical study shows that the proposed RB index policy is nearly optimal across the instances considered, and substantially outperforms several previously proposed index policies.     

Sensible decisions based on QoS

Network Quality of Service (QoS) criteria of interest include conventional metrics such as throughput, delay, loss, and jitter, as well as new QoS criteria based on power utilization, reliability and security. Variable and adaptive routing have again become of interest in networking because of the increasing importance of mobile ad-hoc networks. In this paper we develop a probability model of adaptive routing algorithms which use the expected QoS to select paths in the network. Our objective is not to analyze QoS, but rather to design randomized routing policies which can improve QoS. We define QoS metrics as non-negative random variables associated with network paths which satisfy a sub-additivity condition along each path. We define the QoS of a path, under some routing policy, as the expected value of a non-decreasing measurable function of the QoS metric. We discuss sensitive and insensitive QoS metrics, the latter being dependent on the routing policy which is used. We describe routing policies simply as probabilistic choices among all possible paths from some source to some given destination. Incremental routing policies are defined as those which can be derived from independent decisions taken at certain points (or nodes) along paths. Sensible routing policies are then introduced: they take decisions based simply on the QoS of each available path. Sensible policies, which make decisions based on the QoS of the paths, are introduced. We prove that the routing probability of a sensible policy can always be uniquely obtained. A hierarchy of m-sensible probabilistic routing policies is then introduced. A 0-sensible policy is simply a random choice of routes with equal probability, while a 1-sensible policy selects a path with a probability which is inversely proportional to the (expected) QoS of the path. We prove that an m + 1-sensible policy provides better QoS on the average than an m-sensible policy, if the QoS metric is insensitive. We also show that under certain conditions, the same result also holds for sensitive QoS metrics.Accepted: May 2003, This work was supported by the U.S. Army and Navy under contracts N611339-00-K-0002, N61339-02-C0050, N61339-02-C0080, N61339-02-C0117, and by NSF grants EIA0086251, EIA0203446, ECS0216381.     

Jittering performance of random deflection routing in packet networks

This paper reports a study of random deflection routing protocol and its impact on delay-jitter over packet networks. In case of congestion, routers with a random deflection routing protocol can forward incoming packets to links laying off shortest paths; namely, packets can be “deflected” away from their original paths. However, random deflection routing may send packets to several different paths, thereby introducing packet re-ordering. This could affect the quality of receptions, it could also impair the overall performance in transporting data traffic. Nevertheless, the present study reveals that deflection routing could actually reduce delay-jitter when the traffic loading on the network is not heavy.     

ATM Routing Algorithms for Multimedia Traffic in Private ATM Networks

The goal of this paper is to recommend a good Private Network-to-Network Interface (PNNI) routing algorithm for private ATM networks. A good routing algorithm has to work well with multimedia traffic with several quality of service (QoS) requirements (such as cell loss ratio, cell delay and its variation etc.) in different networks under various traffic conditions. The multiplicity of QoS requirements makes the routing problem NP-complete, so our approach to the problem is based on large scale simulations involving several empirical algorithms (compliant with the PNNI routing specification) which have been tested for different network topologies and traffic scenarios. Based on analysis of tradeoffs involving performance metrics (such as blocking rate, complexity, load distribution) we recommend a consistently good routing algorithm for single domain ATM networks.     

Stochastic Networks: Admission and Routing Using Penalty Functions

We propose an admission and routing control policy for a network of service facilities in a stochastic setting in order to maximize a long run average reward. Queueing and reneging before entering the network is allowed; we introduce orbiting as an approximation to the queueing. Once a customer has entered the network, it incurs no more waiting. Our control policy is easy to implement and we prove that it performs well in steady state as long as the capacity request sizes are relatively small compared to the capacity of the service facilities. The policy is a target tracking policy: a linear program provides a target operating point and an exponential penalty function is used to translate the optimal deterministic point into a feasible admission and routing policy. This translation essentially transforms the admission and routing control problem into a problem of load balancing via the construction of fictitious systems. Simulation studies are included to illustrate that our policy also performs well when request sizes are moderate or large with respect to the capacity.     

An interpolation approximation for the GI/G/1 queue based on multipoint Padé approximation

The performance evaluation of many complex manufacturing, communication and computer systems has been made possible by modeling them as queueing systems. Many approximations used in queueing theory have been drawn from the behavior of queues in light and heavy traffic conditions. In this paper, we propose a new approximation technique, which combines the light and heavy traffic characteristics. This interpolation approximation is based on the theory of multipoint Padé approximation which is applied at two points: light and heavy traffic. We show how this can be applied for estimating the waiting time moments of the ^GI/G/1 queue. The light traffic derivatives of any order can be evaluated using the MacLaurin series analysis procedure. The heavy traffic limits of the ^GI/G/1 queue are well known in the literature. Our technique generalizes the previously developed interpolation approximations and can be used to approximate any order of the waiting time moments. Through numerical examples, we show that the moments of the steady state waiting time can be estimated with extremely high accuracy under all ranges of traffic intensities using low orders of the approximant. We also present a framework for the development of simple analytical approximation formulas. This revised version was published online in June 2006 with corrections to the Cover Date.     

Dynamic Routing in Fully Connected Networks

This paper considers various aspects of dynamic routing in fullyconnected circuit-switched networks. Bounds are obtained forthe performance of any dynamic routing scheme, some theoreticalconsiderations affecting the choice of trunk reservation parametersare presented, and a very simple dynamic routing scheme calleddynamic alternative routing is described, which performs wellunder a variety of conditions. Finally, some of the capacityplanning issues which arise for networks using dynamic routingare discussed.     

Improved bounds for queues with delayed arrivals

Improved bounds are developed for a queue where arrivals are delayed by a fixed time. For moderate to heavy traffic, a simple improved upper bound is obtained which only uses the first two moments of the service time distribution. We show that our approach can be extended to obtain bounds for other types of delayed arrival queues. For very light traffic, asymptotically tight bounds can be obtained using more information about the service time distribution. While an improved upper bound can be obtained for light to moderate traffic it is not particularly easy to apply. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the determination of the control parameters of the optimal can-order policy

This paper considers the well-known class of can-order policies. This type of coordinated replenishment policies accounts for a joint set-up cost structure, where a major set-up cost is incurred for any order and an individual minor set-up cost is charged for each item in the replenishment. Recent comparative studies have pointed out that the performance of the optimal can-order policy is poor, compared to other coordinated replenishment strategies, when the major set-up cost is high. This paper shows that it is the approximate decomposition method to calculate the optimal canorder parameters which performs bad in such situations and not the policy itself. Attention is focused to a subclass of can-order policies, which is close to the optimal can-order policy for high major set-up costs. A solution procedure is developed to calculate the optimal control parameters of this policy. It is shown that a properly chosen combination of the solution procedures to calculate can-order parameters leads to a can-order strategy which performs as well as other coordinated replenishment policies.     

Dynamic Scheduling of a Multiclass Fluid Model with Transient Overload

We study the optimal dynamic scheduling of different requests of service in a multiclass stochastic fluid model that is motivated by recent and emerging computing paradigms for Internet services and applications. In particular, our focus is on environments with specific performance guarantees for each class under a profit model in which revenues are gained when performance guarantees are satisfied and penalties are incurred otherwise. Within the context of the corresponding fluid model, we investigate the dynamic scheduling of different classes of service under conditions where the workload of certain classes may be overloaded for a transient period of time. Specifically, we consider the case with two fluid classes and a single server whose capacity can be shared arbitrarily among the two classes. We assume that the class 1 arrival rate varies with time and the class 1 fluid can more efficiently reduce the holding cost. Under these assumptions, we characterize the optimal server allocation policy that minimizes the holding cost in the fluid model when the arrival rate function for class 1 is known. Using the insights gained from this deterministic case, we study the stochastic fluid system when the arrival rate function for class 1 is random and develop various policies that are optimal or near optimal under various conditions. In particular, we consider two different types of heavy traffic regimes and prove that our proposed policies are strongly asymptotically optimal. Numerical examples are also provided to demonstrate further that these policies yield good results in terms of minimizing the expected holding cost.     

主路多车道无信号交叉口混合交通流的通行能力

Highway capacity is defined as maximum volume of traffic flow through the particular highway section under given traffic conditions,road conditions and so on.Highway construction and management is judged by capacity standard.The reasonable scale and time of highway construction,rational network structure and optimal management mode of highway network can be determined by analyzing the fitness between capacity and traffic volume.All over the world,highway capacity is studied to different extent in different country. Based on the gap acceptance theory,the mixed traffic flow composed of two representative vehicle types heavy and light vehicles is analyzed with probability theory.Capacity model of the minor mixed traffic flows crossing m major lanes,on which the traffic flows fix in with M3 distributed headway,on the unsignalized intersection is set up,and it is an extension of minor lane capacity theory for one vehicle-type and one major-lane traffic flow.     

Waiting time based routing policies to parallel queues with percentiles objectives

We develop a method to obtain near-optimal routing policies to parallel queues with decisions based on customers’ wait and performance objectives which include percentiles of the waiting time. We formulate and explicitly derive a value function where the waiting time is used as a decision variable. This allows us to apply a one-step policy improvement method to obtain an efficient routing solution. Numerical illustrations reveal that classical monotone policies are not always optimal.     

Scheduling a repairman in a finite source system

The scheduling of a single server in a finite source model is considered. TheN customers in the system have different failure and repair rates. Also the costs depend on the customers which are broken down. We give a condition under which the average costs are minimized by a simple list policy, and with a counterexample we show that in the general case no optimal list policy may exist. This motivates us to derive policies which are optimal under low and high traffic conditions. They are again list policies, which behave well numerically.     

Optimal control in light traffic Markov decision processes

We consider Markov Decision Processes under light traffic conditions. We develop an algorithm to obtain asymptotically optimal policies for both the total discounted and the average cost criterion. This gives a general framework for several light traffic results in the literature. We illustrate the method by deriving the asymptotically optimal control of a simple ATM network.