Routing in multi-class queueing networks

PhD ThesisWe consider the problem of routing (incorporating local scheduling) in a distributed network. Dedicated jobs arrive directly at their specified station for processing. The choice of station for generic jobs is open. Each job class has an associated holding cost rate. We aim to develop routing policies to minimise the long-run average holding cost rate. We first consider the class of static policies. Dacre, Glazebrook and Nifio-Mora (1999) developed an approach to the formulation of static routing policies, in which the work at each station is scheduled optimally, using the achievable region approach. The achievable region approach attempts to solve stochastic optimisation problems by characterising the space of all possible performances and optimising the performance objective over this space. Optimal local scheduling takes the form of a priority policy. Such static routing policies distribute the generic traffic to the stations via a simple Bernoulli routing mechanism. We provide an overview of the achievements made in following this approach to static routing. In the course of this discussion we expand upon the study of Becker et al. (2000) in which they considered routing to a collection of stations specialised in processing certain job classes and we consider how the composition of the available stations affects the system performance for this particular problem. We conclude our examination of static routing policies with an investigation into a network design problem in which the number of stations available for processing remains to be determined. The second class of policies of interest is the class of dynamic policies. General DP theory asserts the existence of a deterministic, stationary and Markov optimal dynamic policy. However, a full DP solution may be unobtainable and theoretical difficulties posed by simple routing problems suggest that a closed form optimal policy may not be available. This motivates a requirement for good heuristic policies. We consider two approaches to the development of dynamic routing heuristics. We develop an idea proposed, in the context of simple single class systems, by Krishnan (1987) by applying a single policy improvement step to some given static policy. The resulting dynamic policy is shown to be of simple structure and easily computable. We include an investigation into the comparative performance of the dynamic policy with a number of competitor policies and of the performance of the heuristic as the number of stations in the network changes. In our second approach the generic traffic may only access processing when the station has been cleared of all (higher priority) jobs and can be considered as background work. We deploy a prescription of Whittle (1988) developed for RBPs to develop a suitable approach to station indexation. Taking an approximative approach to Whittle's proposal results in a very simple form of index policy for routing the generic traffic. We investigate the closeness to optimality of the index policy and compare the performance of both of the dynamic routing policies developed here

Index heuristics for routing and service control problems within queueing systems

This thesis is naturally broken down into two main problems, one concerning optimal routing control and the other optimal service control. In the routing control problem the arriving customers must be allocated to one of the 'K' possible service stations. We assume that the customers arrive in a single Poisson stream. We take the service at each of the stations to be exponentially distributed, but perhaps with different parameters. The system cost rate is additive across the queues formed at each station. We also have that at each station the holding cost function is increasing convex. Following Whittle's approach to a class of restless bandit problems, we develop a Lagrangian relaxation of the routing control problem which serves to motivate the development of index heuristics. The index by a particular station is characterised as a fair charge for rejecting the arriving customer at that station. We also consider a policy improvement index for comparison to the heuristic. We develop these indices and report an extensive numerical investigation which exhibits strong performance of the index heuristic for both discounted and average costs.The second problem concerns the optimal service control of a multi-class M/G/l queueing system in which customers are served non preemptively. The system cost rate is additive across classes and increasing convex in the numbers present within each class. We again follow the method prescribed by Whittle when considering a class of restless bandits. Hence we develop a Lagrangian relaxation of the service control problem which motivates the development of a class of index heuristics. For a particular customer class the index is characterised as a fair charge for service of that class. These indices are developed and we again report representative results from an extensive numerical study which again implies a strong performance of the index heuristic for both discounted and average costs

Outsourcing warranty repairs: models for the allocation of failed items to multiple vendors

We consider a scenario in which several external service vendors are contracted to repair purchased items which fail under warranty. We develop and analyze various allocation models concerning how the repair work should be distributed among the vendors in a cost-effective manner. Furthermore, we depart from previous work by arguing the importance of approaches to the modelling of goodwill costs which penalize long waits experienced by individual customers.We firstly study a simple static allocation model with a fixed warranty population. Both theoretical considerations and numerical results show that a simple greedy approach to the distribution of items under static models works outstandingly well. However, such a static formulation ignores the stochastic nature of the warranty population. Hence, we develop a second allocation model in which new equipment purchases are made according to a compound Poisson process. As in the static allocation model, the current information regarding the repair queue at each vendor is not available to the decision maker. The resulting stochastic dynamic optimization problem is non-standard. We develop an effective allocation procedure to this non-standard problem using a dynamic programming policy improvement approach. We report representative results from a simulation investigation to evaluate the status of the allocation heuristic developed in comparison to two simpler heuristics suggested by static models. Thirdly, we propose a dynamic allocation model which utilizes data on the queue length at each vendor for decisions on the routing of real-time failures to the vendors. Due to the problem size and state space in practice, traditional stochastic dynamic programming is not a realistic and computationally viable option. Hence, Whittle's restless bandits approach is deployed to develop the index-based heuristic for this dynamic allocation problem. A crucial theoretical result in this part of the study is that the system considered is indeed indexable. All the numerical results reported show that the performance of the derived index policy from the restless bandit is superior to that of a range of alternatives

Scheduling a Make-To-Stock Queue: Index Policies and Hedging Points

A single machine produces several different classes of items in a make-to-stock mode. We consider the problem of scheduling the machine to regulate finished goods inventory, minimizing holding and backorder or holding and lost sales costs. Demands are Poisson, service times are exponentially distributed, and there are no delays or costs associated with switching products. A scheduling policy dictates whether the machine is idle or busy, and specifies the job class to serve in the latter case. Since the optimal solution can only be numerically computed for problems with several products, our goal is to develop effective policies that are computationally tractable for a large number of products. We develop index policies to decide which class to serve, including Whittle's "restless bandit" index, which possesses a certain asymptotic optimality. Several idleness policies, which are characterized by hedging points, are derived, and the best policy is obtained from a heavy traffic diffusion approximation. Nine sample problems are considered in a numerical study, and the average suboptimality of the best policy is less than 3%

The scheduling of queues with non-linear holding costs

PhD ThesisWe consider multi-class, single server queueing systems and we seek to devise policies for server allocation which minimise some long-term cost function. In most of the work to date on the optimal dynamic control of such systems, holding cost rates are assumed to be linear in the number of customers present. Such assumptions have been argued to be unrealistic and thus inappropriate: see Van Meighem (1995). With pure priority policies, which often emerge from analyses based on linear holding cost assumptions, there is often the problem that service offered to lower priority traffic is unacceptably poor. Seeking to address such problems, we first investigate the performance of policies based on linear switching curves in an M/M/1 model with two customer types, imposing various constraints on the second moments of queue lengths. We then develop an index heuristic for a multi-class M/M/1 model with increasing convex holding cost rates. Following work by Whittle (1988), we develop the required indices and in a numerical study of two and three class systems, demonstrate the strong performance of these index policies. Performance of policies throughout the thesis, as measured by lowest costs achievable under a given policy class, (i. e. best linear switching, best threshold, or index policy) is compared with a lower bound on the minimum cost achievable under any policy. This lower bound is obtained by adopting the achievable region approach, see Bertsimas, Paschalidis & Tsitsiklis (1994) and Bertsimas & Nino-Mora (1996) in which we construct a set of constraints satisfied by the first and second moments of the queue lengths. These constraints define a relaxation of the set of achievable region performance vectors of the system. Optimisation over this relaxed region yields the lower bound. Numerical results indicate the strong performance of the index policy.Engineering and Physical Sciences Research Council

The scheduling of queues with non-linear holding costs

We consider multi-class, single server queueing systems and we seek to devise policies for server allocation which minimise some long-term cost function. In most of the work to date on the optimal dynamic control of such systems, holding cost rates are assumed to be linear in the number of customers present. Such assumptions have been argued to be unrealistic and thus inappropriate: see Van Meighem (1995). With pure priority policies, which often emerge from analyses based on linear holding cost assumptions, there is often the problem that service offered to lower priority traffic is unacceptably poor. Seeking to address such problems, we first investigate the performance of policies based on linear switching curves in an M/M/1 model with two customer types, imposing various constraints on the second moments of queue lengths. We then develop an index heuristic for a multi-class M/M/1 model with increasing convex holding cost rates. Following work by Whittle (1988), we develop the required indices and in a numerical study of two and three class systems, demonstrate the strong performance of these index policies. Performance of policies throughout the thesis, as measured by lowest costs achievable under a given policy class, (i. e. best linear switching, best threshold, or index policy) is compared with a lower bound on the minimum cost achievable under any policy. This lower bound is obtained by adopting the achievable region approach, see Bertsimas, Paschalidis ; Tsitsiklis (1994) and Bertsimas ; Nino-Mora (1996) in which we construct a set of constraints satisfied by the first and second moments of the queue lengths. These constraints define a relaxation of the set of achievable region performance vectors of the system. Optimisation over this relaxed region yields the lower bound. Numerical results indicate the strong performance of the index policy.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research CouncilGBUnited Kingdo

Time-dependent performance approximation of truck handling operations at an air cargo terminal

This paper provides an analytical solution for the time-dependent performance evaluation of truck handling operations at an air cargo terminal. The demand for loading and unloading operations is highly time-dependent and stochastic for two classes of trucks. Two heterogeneous handling facilities with multiple servers are available to handle trucks assuming exponentially distributed processing times. Trucks are routed to a handling facility depending on the current state of the system upon arrival. To approximate the time-dependent behavior of such heterogeneous queueing systems, we develop a stationary backlog-carryover (SBC) approach. A numerical study compares this approach with simulations and demonstrates its applicability to real-world input data

Routing and transfers amongst parallel queues

This thesis is concerned with maximizing the performance of policies for routing and transferring jobs in systems of heterogeneous servers. The tools used are probabilistic modelling, optimization and simulation. First, a system is studied where incoming jobs are allocated to the queue belonging to one of a number of servers, each of which goes through alternating periods of being operative and inoperative. The objective is to evaluate and optimize performance and cost metrics. Jobs incur costs for the amount of time that they spend in a queue, before commencing service. The optimal routing policy for incoming jobs is obtained by solving numerical programming equations. A number of heuristic policies are compared against the optimal, and one dynamic routing policy is shown to perform well over a large range of parameters. Next, the problem of how best to deal with the transfer of jobs is considered. Jobs arrive externally into the queue attached to one of a number of servers, and on arrival are assigned a time-out period. Jobs whose time-out period expires before it commences service is instantaneously transferred to the end another queue, based on a routing policy. Upon transfer, a transfer cost is incurred. An approximation to the optimal routing policy is computed, and compared with a number of heuristic policies. One heuristic policy is found to perform well over a large range of parameters. The last model considered is the case where incoming jobs are allocated to the queue attached to one of a number of servers, each of which goes through periods of being operative and inoperative. Additionally, each job is assigned a time-out on arrival into a queue. Any job whose time-out period expires before it commences service is instantaneously transferred to the end of another queue, based on a transfer policy. The objective is to evaluate and optimize performance and cost metrics. Jobs incur costs for the amount of time that they spend in a queue, before commencing service, and additionally incur a cost for each transfer they experience. A number of heuristic transfer policies are evaluated and one heuristic which performs for a wide range of parameters is observed.EThOS - Electronic Theses Online ServiceGBUnited Kingdo