Nonidling Policies Research Articles

Increasing social welfare with delays: Strategic customers in the M/G/1 orbit queue

Strategic customers typically patronize service systems at a higher rate than the socially optimal one. Much literature has focused on inducing customers to join such systems at this latter rate. This entire literature considers nonidling policies that are the focus of queueing theory. We demonstrate that strategically imposing delays into service systems can improve the social welfare using the M/G/1 queue with orbit. This versatile queueing model has been extensively studied from a performance evaluation perspective. In this system, customers who arrive and find the server idle begin immediately their service. However, strategic customers who find the server busy decide whether to balk or join a virtual queue, that is, an orbit. Then, each time the server finishes a service, he begins to retrieve a customer from the orbit and the corresponding retrieving time is not negligible. These retrieving times function as extra delays that are imposed on customers that find a busy server. We show that when customers are strategic, there are certain ranges of the parameters where delaying the orbit customers can increase the welfare of a system or even maximize it. To this end, we characterize and compute the equilibrium strategies for the customers' joining/balking dilemma. We consider both the unobservable and observable versions of the system, and provide some insight on the optimal delay and level of information in such systems. We further show that the welfare for this system is higher than the corresponding standard M/G/1 queue with the same delay.

Increasing social welfare with delays: strategic customers in the M/G/1 orbit queue

It is well known that strategic customers typically patronize service systems at a higher rate than the socially optimal one. Much literature has focused on inducing customers to patronize such systems at this latter rate. All of this literature considered non-idling policies that are the focus of queueing theory. We demonstrate that strategically imposing delays into such systems can improve the social welfare using the M/G/1 queue with orbit. This queue is a versatile model, that has been extensively studied in the literature from a performance evaluation perspective. In this system, customers who arrive and find the server idle begin immediately their service. However, strategic customers who find the server busy can decide whether to balk or join a virtual queue that is referred to as the orbit. Then, each time the server finishes a service, he begins to retrieve a customer from the orbit and the corresponding retrieving time is not negligible. These retrieving times can be seen as extra delays that are imposed on customers that find a busy server. We further show that when customers are strategic there are certain ranges of the parameters where strategically delaying the orbit customers can increase the welfare of a system or even maximize it. To this end, we characterize and compute the equilibrium strategies for the customers' joining/balking dilemma. We consider both the unobservable and observable versions of the system, compare them and provide some insight on the optimal delay and level of information in such systems.

Performance evaluation of scheduling policies for the dynamic and stochastic resource-constrained multi-project scheduling problem

In this study, we consider the dynamic and stochastic resource-constrained multi-project scheduling problem where projects generate rewards at their completion, completions later than a due date cause tardiness costs, task duration is uncertain, and new projects arrive randomly during the ongoing project execution both of which disturb the existing project scheduling plan. We model this problem as a discrete-time Markov decision process and explore the performance and computational limitations of solving the problem by dynamic programming. We run and compare five different solution approaches, which are: a dynamic programming algorithm to determine a policy that maximises the time-average profit, a genetic algorithm and an optimal reactive baseline algorithm, both generate a schedule to maximise the total profit of ongoing projects, a rule-based algorithm which prioritises processing of tasks with the highest processing durations, and a worst decision algorithm to seek a non-idling policy that minimises the time-average profit. The performance of the optimal reactive baseline algorithm is the closest to the optimal policies of the dynamic programming algorithm, but its results are suboptimal, up to 37.6%. Alternative scheduling algorithms are close to optimal with low project arrival probability but quickly deteriorate their performance as the probability increases.

Optimality of the fastest available server policy

We give sufficient conditions under which a policy that assigns customers to the Fastest Available Server, labeled FAS, is optimal among nonidling policies in queueing models with multiple independent Markov-modulated Poisson arrival processes and heterogeneous parallel exponential servers with server-dependent service rates. The criteria are to minimize the long-run average cost per unit time and the expected present value of the costs. We obtain results for loss and delay queueing models with finite- or infinite-capacity buffers under a first-come-first-served priority scheme. We analyze the robustness of the cost structure assumptions that are invoked in the proof of FAS optimality.

Optimality of the Fastest Available Server Policy

We give sufficient conditions under which a policy that assigns customers to the Fastest Available Server, labelled FAS, is optimal among non-idling policies in queuing models with multiple independent Markov-modulated Poisson arrival processes and heterogeneous parallel exponential servers with server-dependent service rates. The criteria are to minimize the long-run average cost per unit time and the expected present value of the costs. We obtain results for loss and delay queuing models with finite- or infinite-capacity buffers under a first-come-first serve priority scheme. We analyze the robustness of the cost structure assumptions that are invoked in the proof of FAS optimality.

Dynamic Server Assignment With Task-Dependent Server Synergy

We study tandem queueing systems with finite buffers in which servers work more efficiently in teams than on their own and the synergy among collaborating servers can be task-dependent. Our goal is to determine the dynamic server assignment policy that maximizes the long-run average throughput. When each server works with the same ability at each task that she/he is assigned to, we show that any nonidling policy where all servers work in teams of two or more at all times is optimal. On the other hand, when the server abilities are task-dependent, we show that for Markovian systems with two stations and two servers, depending on the synergy among the servers, the optimal policy either assigns the two servers to different stations when possible, or lets them work in a team at all times. Finally, for larger Markovian systems, we provide sufficient conditions that guarantee that the optimal policy has all servers working together at all times.

Optimal control of noncollaborative servers in two‐stage tandem queueing systems

AbstractWe consider two‐stage tandem queueing systems with dedicated servers in each station and a flexible server that is trained to serve both stations. We assume no arrivals, exponential service times, and linear holding costs for jobs present in the system. We study the optimal dynamic assignment of servers to jobs assuming a noncollaborative work discipline with idling and preemptions allowed. For larger holding costs in the first station, we show that (i) nonidling policies are optimal and (ii) if the flexible server is not faster than the dedicated servers, the optimal server allocation strategy has a threshold‐type structure. For all other cases, we provide numerical results that support the optimality of threshold‐type policies. Our numerical experiments also indicate that when the flexible server is faster than the dedicated server of the second station, the optimal policy may have counterintuitive properties, which is not the case when a collaborative service discipline is assumed. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 435–446, 2014

Queue-Length Asymptotics for Generalized Max-Weight Scheduling in the Presence of Heavy-Tailed Traffic

We investigate the asymptotic behavior of the steady-state queue-length distribution under generalized max-weight scheduling in the presence of heavy-tailed traffic. We consider a system consisting of two parallel queues, served by a single server. One of the queues receives heavy-tailed traffic, and the other receives light-tailed traffic. We study the class of throughput-optimal max-weight- $\alpha $ scheduling policies and derive an exact asymptotic characterization of the steady-state queue-length distributions. In particular, we show that the tail of the light queue distribution is at least as heavy as a power-law curve, whose tail coefficient we obtain explicitly. Our asymptotic characterization also shows that the celebrated max-weight scheduling policy leads to the worst possible tail coefficient of the light queue distribution, among all nonidling policies. Motivated by the above negative result regarding the max-weight- $\alpha $ policy, we analyze a log-max-weight (LMW) scheduling policy. We show that the LMW policy guarantees an exponentially decaying light queue tail while still being throughput-optimal.

Optimal scheduling of parallel queues using stochastic flow models

We consider a classic scheduling problem for optimally allocating a resource to multiple competing users and place it in the framework of Stochastic Flow Models (SFMs). We derive Infinitesimal Perturbation Analysis (IPA) gradient estimators for the average holding cost with respect to resource allocation parameters. These estimators are easily obtained from a sample path of the system without any knowledge of the underlying stochastic process characteristics. Exploiting monotonicity properties of these IPA estimators, we prove the optimality of the well-known cμ-rule for an arbitrary finite number of queues and stochastic processes under non-idling policies and linear holding costs. Further, using the IPA derivative estimates obtained along with a gradient-based optimization algorithm, we find optimal solutions to similar problems with nonlinear holding costs extending current results in the literature.

Optimal Scheduling of Parallel Queues with Stochastic Flow Models: The cμ-rule Revisited

AbstractWe consider a classic scheduling problem for optimally allocating a resource to multiple competing users and place it in the framework of Stochastic Flow Models (SFMs). We derive Infinitesimal Perturbation Analysis (IPA) gradient estimators for the average holding cost with respect to resource allocation parameters. These estimators are easily obtained from a sample path of the system without any knowledge of the underlying stochastic process characteristics. Exploiting monotonicity properties of these IPA estimators, we prove the optimality of the well-known cμ-rule not only for the two-queue case (as in earlier work) but for an arbitrary finite number of queues and stochastic processes under non-idling policies.

MAXIMIZING THE THROUGHPUT OF TANDEM LINES WITH FLEXIBLE FAILURE-PRONE SERVERS AND FINITE BUFFERS

Consider a tandem queuing network with an infinite supply of jobs in front of the first station, infinite room for completed jobs after the last station, finite buffers between stations, and a number of flexible servers who are subject to failures. We study the dynamic assignment of servers to stations with the goal of maximizing the long-run average throughput. Our main conclusion is that the presence of server failures does not have a major impact on the optimal assignment of servers to stations for the systems we consider. More specifically, we show that when the servers are generalists, any nonidling policy is optimal, irrespective of the reliability of the servers. We also provide theoretical and numerical results for Markovian systems with two stations and two or three servers that suggest that the structure of the optimal server assignment policy does not depend on the reliability of the servers and that ignoring server failures when assigning servers to stations yields near-optimal throughput. Finally, we present numerical results that illustrate that simple server assignment heuristics designed for larger systems with reliable servers also yield good throughput performance in Markovian systems with three stations and three failure-prone servers.

OPTIMAL CONTROL OF FLEXIBLE SERVERS IN TWO TANDEM QUEUES WITH OPERATING COSTS

We consider two-stage tandem queuing systems with dedicated servers in each station and flexible servers that can serve in both stations. We assume exponential service times, linear holding costs, and operating costs incurred by the servers at rates proportional to their speeds. Under conditions that ensure the optimality of nonidling policies, we show that the optimal allocation of flexible servers is determined by a transition-monotone policy. Moreover, we present conditions under which the optimal policy can be explicitly determined.

Instability in stochastic and fluid queueing networks

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all nonidling policies. However, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations, we prove that if the fluid model is not weakly stable under the class of all nonidling policies, then a corresponding queueing network is not rate stable under the class of all nonidling policies. We establish the result by building a particular nonidling scheduling policy which makes the associated stochastic process transient. An important corollary of our result is that the condition ρ*≤1, which was proven in [Oper. Res. 48 (2000) 721–744] to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for an associated queueing network. Here ρ* is a certain computable parameter of the network involving virtual station and push start conditions.

An Intuitive Condition for Stability and Performance Analysis of Multiclass Queueing Networks

We present an intuitive stability condition for open multiclass queueing networks with Bernoulli routing: if each station has enough service capacity to cope with its peak traffic intensity, then the network is stable under any stationary nonidling scheduling policy. The condition is close to sharp for networks with light traffic between stations. Under this peak-rate condition, in the case of Markovian networks, we derive a closed-form upper bound on the time-average number of customers in the system, which is uniformly valid under all stationary nonidling policies. Our proof combines two recent results concerning (1) the relation between stability and performance via linear programming developed by Kumar and Meyn (1996); and (2) the work decomposition laws for multiclass queueing networks of Bertsimas and Ni?o-Mora (1999). The stability condition is tested on a generalization of the Lu-Kumar network, which shows how its quality depends on the degree of network connectivity.

On the guaranteed throughput and efficiency of closed re-entrant lines

A closed network is said to be “guaranteed efficient” if the throughput converges under all non-idling policies to the capacity of the bottlenecks in the network, as the number of trapped customers increases to infinity. We obtain a necessary condition for guaranteed efficiency of closed re-entrant lines. For balanced two-station systems, this necessary condition is almost sufficient, differing from it only by the strictness of an inequality. This near characterization is obtained by studying a special type of virtual station called “alternating visit virtual station”. These special virtual stations allow us to relate the necessary condition to certain indices arising in heavy traffic studies using a Brownian network approximation, as well as to certain policies proposed as being extremal with respect to the asymptotic loss in the throughput. Using the near characterization of guaranteed efficiency we also answer the often pondered question of whether an open network or its closed counterpart has greater throughput – the answer is that neither can assure a greater guaranteed throughput.

The Throughput of Irreducible Closed Markovian Queueing Networks: Functional Bounds, Asymptotic Loss, Efficiency, and the Harrison-Wein Conjectures

Let N be the population of an irreducible closed Markovian queueing network, and denote by αu(N) the throughput of a scheduling policy u. The policy u is said to be efficient if limN→+∞αu(N) = α*, where α* is the capacity of a bottleneck station. The quantity J(u) ≔ limN→∞ N(α* − αu(N))/α* is called the asymptotic loss of u. The policy u is said to be asymptotically optimal if J(u) is as small as it can be. For multi-station irreducible closed Markovian networks, we obtain functional bounds on the throughput which are of the form αN/(N + v). The coefficients α and v are obtained by solving two linear programs (LPs), which consist of L(L + 3)/2 variables, where L is the number of buffers in the system. We are thus able to establish efficiency, as well as provide upper and lower bounds on asymptotic loss, through LP procedures. For balanced systems where all stations are equally loaded, we are able to provide reduced dimensional LPs which consist of just S(S + 1)/2 variables, where S is the number of stations in the system. We also show that the lower bound on asymptotic loss produced by the reduced dimensional LP can capture interactions between multiple bottlenecks in heavy traffic, and that it always dominates a bound obtainable by extending a conjecture of Harrison and Wein (HW) from two station systems to multi-station systems. Employing these bounds on two station systems, we prove that a certain policy developed by HW is efficient. This buffer priority policy was conjectured by them to be asymptotically optimal. They also conjectured a finite valued formula for the asymptotic loss of every buffer priority policy. This latter conjecture is not true in its full generality since there exists a certain buffer priority policy for a certain two station system which is not efficient, and consequently has infinite asymptotic loss. However, we provide credence to the conjectures by establishing that for balanced systems, the conjectured asymptotic loss of the HW-policy is a lower bound on the asymptotic loss of all policies, while the conjectured formula for the asymptotic loss applied to the exact opposite of the HW-policy is an upper bound on the asymptotic loss for all nonidling policies. The latter result is established under an additional condition identified by us which guarantees the finiteness of the asymptotic loss, and thus also the efficiency, of every nonidling scheduling policy.

Optimal Server Allocation to Parallel Queues with Finite-Capacity Buffers

In this paper, we study the problem of dynamic allocation of a single server to parallel queues with finite-capacity buffers. The arrival processes are mutually independent, equal rate Poisson processes, and the service times are independent and identically distributed random variables with an arbitrary distribution. We are interested in characterizing the allocation policy that stochastically minimizes the number of customers lost due to buffer overflows. Using a coupling argument, we establish the optimality of the Fewest-Empty-Spaces policy, which allocates the server to the queue with the fewest empty buffer spaces, within the class of nonpreemptive and nonidling policies. The result extends to the class of preemptive policies, if the service times are exponentially distributed. We also briefly discuss the allocation problem under more general statistical assumptions on the arrival processes.

Optimal intensity allocation of single-server queueing networks

Consider an optimal intensity allocation problem of a non-markovian queueing network with a single server. Both the arrival and service processes are modelled by point processes with stochastic intensities. All the stations share a single service facility. After finishing service at one station, a customer switches to another station or leaves the system according to specific probabilities. A linear holding cost is incurred at each station. The objective is to control the arrival intensity at each station and schedule the service intensity at all stations so that the expected discounted cost over an infinite horizon is minimized. The main result is that in the class of non-idling control policies, the optimal intensity is of a bang-bang type and the optimal scheduling rule turns out to be the Klimov's rule; whereas in the larger class of control policies that permits enforced idling, the optimal control rule is a modified static policy. The approach combines a point process calculus with dynamic programming

Stability and performance of distributed production control methods based on continuous-flow models

Investigates the stability and performance of a two-level production control method for part release, routing, and machine scheduling in manufacturing systems. At the first level (off line), a continuous-flow (fluid) approximation of the production control problem is formulated and solved as a linear program. At the second level (on line), simple distributed control policies are used to track the solution of the continuous-flow model of level one. A major research issue concerns the potential instability of decentralized tracking policies due to the discreteness of the decision space (caused by the sequential nature of operations and their discrete processing times). The author considers a broad class of distributed tracking policies, which is called nonidling-nonexceeding (NINE) and finds a sufficient condition for their stability. The results show the NINE policies potential instability for nonacyclic systems due to machine starvation, which is caused by tracking delays induced by the discrete nature of operations. The sufficient stability condition takes a familiar form, namely that, the largest eigenvalue of a matrix/spl minus/which captures the dynamics of tracking delays in the system/spl minus/must be less than one. It ensures the contraction of tracking delays in the feedback loops of material flow in the system. When this condition is met, an upper bound on the performance of the policy is readily obtained. It is further shown that any nonidling dispatching policy can be considered as a special NINE policy, and thus the same stability and performance results apply to the class of all nonidling policies. The author also investigates NINE policies with buffer priorities and shows that a NINE policy with certain buffer priority ordering is always stable. Simulation experiments show that near-zero work-in-process and finished-parts' inventory can be achieved with the method even for demands that are very close to the production capacity of the system. >

Stochastic Scheduling in in-Forest Networks

In this paper we study the extremal properties of several scheduling policies in an in-forest network consisting of multiserver queues. Each customer has a due date, and we assume that service times at the different queues form mutually independent sequences of independent and identically distributed random variables independent of the arrival times and due dates. Furthermore, the network is assumed to consist of a mixture of nodes, some of which permit only non-preemptive service policies whereas the others permit preemptive resume policies. In the case of non-preemptive queues, service times may be generally distributed if there is only one server; otherwise the service times are required to be increasing in likelihood ratio (ILR). In the case of preemptive queues, service times are restricted to exponential distributions. Using stochastic majorizations and partial orders on permutations, we establish that, within the class of work conserving service policies, the stochastically smallest due date (SSDD) and the stochastically largest due date (SLDD) policies minimize and maximize, respectively, the vector of the customer latenesses of the first n customers in the sense of the Schur-convex order and some weaker orders, provided the due dates are comparable in some stochastic sense. It then follows that the first come-first served (FCFS) and the last come-first served (LCFS) policies minimize and maximize, respectively, the vector of the response times of the first n customers in the sense of the Schur-convex order. We also show that the FCFS and LCFS policies minimize and maximize, respectively, the vector of customer end-to-end delays in the sense of the strong stochastic order. Extensions to the class of non-idling policies and to the stationary regime are also given.