Waiting times in polling systems with various service disciplines

We consider a polling system of N queues Q1,..., QN, cyclically visited by a single server. Customers arrive at these queues according to independent Poisson processes, requiring generally distributed service times. When the server visits Qi, i = 1,..., N, it serves a number of customers according to a certain visit discipline. This discipline is assumed to belong to the class of branching-type disciplines, which includes gated and exhaustive service. The special feature of our study is that, within each queue, we do not restrict ourselves to service in order of arrival (FCFS); we are interested in the effect of different service disciplines, like Last-Come-First-Served, Processor Sharing, Random Order of Service, and Shortest Job First. After a discussion of the joint distribution of the numbers of customers at each queue at visit epochs of the server to a particular queue, we determine the Laplace-Stieltjes transform of the cycle-time distribution, viz., the time between two successive visits of the server to, say, Q1. This yields the transform of the joint distribution of past and residual cycle time, w.r.t. the arrival of a tagged customer at Q1. Subsequently concentrating on the case of gated service at Q1, we use that cycle-time result to determine the (Laplace-Stieltjes transform of the) waiting-time distribution at Q1. Next to locally gated visit disciplines, we also consider the globally gated discipline. Again, we consider various non-FCFS service disciplines at the queues, and we determine the (Laplace-Stieltjes transform of the) waiting-time distribution at an arbitrary queue.

Heavy-traffic limits for Polling Models with Exhaustive Service and non-FCFS Service Order Policies

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS local service orders, namely Last-Come-First-Served (LCFS) with and without preemption, Random-Order-of-Service (ROS), Processor Sharing (PS), the multi-class priority scheduling with and without preemption, Shortest-Job-First (SJF) and the Shortest Remaining Processing Time (SRPT) policy. For each of these policies, we rst express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-trac assumptions, i.e., when the system tends to saturate. The results show that in all cases the asymptotic wait

Inventory control in multi-item production systems

This thesis focusses on the analysis and construction of control policies in multiitem production systems. In such systems, multiple items can be made to stock, but they have to share the finite capacity of a single machine. This machine can only produce one unit at a time and if it is set-up for one item, a switch-over or set-up time is needed to start the production of another item. Customers arrive to the system according to (compound) Poisson processes and if they see no stock upon arrival, they are either considered as a lost sale or backlogged. In this thesis, we look at production systems with backlog and production systems with lost sales. In production systems with lost sales, all arriving customers are considered lost if no stock is available and penalty costs are paid per lost customer. In production systems with backlog, arriving customers form a queue if they see no stock and backlogging costs are paid for every backlogged customer per time unit. These production systems find many applications in industry, for instance glass and paper production or bulk production of beers, see Anupindi and Tayur [2]. The objective for the production manager is to minimize the sum of the holding and penalty or backlogging costs. At each decision moment, the manager has to decide whether to switch to another product type, to produce another unit of the type that is set-up or to idle the machine. In order to minimize the total costs, a balance must be found between a fast switching scheme that is able to react to sudden changes in demand and a production plan with a little loss of capacity. Unfortunately, a fast switching scheme results in a loss of capacity, because switching from one product type to another requires a switch-over or set-up time. In the optimal production strategy, decisions depend on the complete state of the system. Because the processes at the different product flows depend on these decisions, the processes also depend on the complete state of the system. This means that the processes at the different product flows are not independent, which makes the analysis and construction of the optimal production strategy very complex. In fact, the complexity of the determination of this policy grows exponentially in the number of product types and if this number is too large, the optimal policy becomes intractable. Production strategies in which decisions depend on the complete system are defined as global lot sizing policies and are often difficult to construct or analyse, because of the dependence between the different product flows. However, in this thesis the construction of a global lot sizing policy is presented which also works for production systems with a large number of product types. The key factor that makes the construction possible is the fact that it is based on a fixed cycle policy. In Chapter 2, the fixed cycle policy is analysed for production systems with lost sales and in Chapter 6, the fixed cycle policy is analysed for production systems with backlog. The fixed cycle policy can be analysed per product flow and this decomposition property allows for the determination of the so called relative values. If it is assumed that one continues with a fixed cycle control, the relative values per product type represent the relative expected future costs for each decision. Based on these relative values, an improvement step (see Norman [65]) is performed which results in a ‘one step improvement’ policy. This policy is constructed and analysed in Chapters 2 and 7 for production systems with lost sales and production systems with backlog, respectively. This global lot sizing policy turns out to perform well compared to other, heuristic production strategies, especially in systems with a high load and demand processes with a high variability. A similar approach as for the production system with a single machine is performed in a system with two machines and lost sales in Chapter 3. Results show that in some cases the constructed strategy works well, although in some systems two separate one step improvement policies perform better. Examples of more heuristic production strategies are gated and exhaustive basestock policies. In these ’local lot sizing‘ policies, decisions depend only on the stock level of the product type that is set-up. But even in these policies, the processes at the different product flows are dependent. This makes the analysis difficult, but for production systems with backlog a translation can be made to a queueing system by looking at the number of products short to the base-stock level. So the machine becomes a server and each product flow becomes a queue. In these queueing systems, also known as polling systems, gated and exhaustive base-stock policies become gated and exhaustive visit disciplines. For polling systems, an exact analysis of the queue length or waiting time distribution is often possible via generating functions or Laplace-Stieltjes transforms. In Chapter 5, the determination of the sojourn time distribution of customers in a polling system with a (globally) gated visit discipline is presented, which comes down to the determination of the lead time distribution in the corresponding production system

EUROPEAN CONFERENCE ON QUEUEING THEORY 2016

International audienceThis booklet contains the proceedings of the second European Conference in Queueing Theory (ECQT) that was held from the 18th to the 20th of July 2016 at the engineering school ENSEEIHT, Toulouse, France. ECQT is a biannual event where scientists and technicians in queueing theory and related areas get together to promote research, encourage interaction and exchange ideas. The spirit of the conference is to be a queueing event organized from within Europe, but open to participants from all over the world. The technical program of the 2016 edition consisted of 112 presentations organized in 29 sessions covering all trends in queueing theory, including the development of the theory, methodology advances, computational aspects and applications. Another exciting feature of ECQT2016 was the institution of the Takács Award for outstanding PhD thesis on "Queueing Theory and its Applications"

Critical fluid limit of a gated processor sharing queue

We consider a sequence of single-server queueing models operating under a service policy that incorporates batches into processor sharing: arriving jobs build up behind a gate while waiting to begin service, while jobs in front of the gate are served according to processor sharing. When they have been completed, the waiting jobs move in front of the gate and the cycle repeats. We model this system with a pair of measure valued processes describing the jobs in front of and behind the gate. Under mild asymptotically critical conditions and a law-of-large-numbers scaling, we prove that the pair of measure-valued processes converges in distribution to an easily described limit, which has an interesting periodic dynamics