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.

The analysis of batch sojourn-times in polling systems

We consider a cyclic polling system with general service times, general switch-over times, and simultaneous batch arrivals. This means that at an arrival epoch, a batch of customers may arrive simultaneously at the different queues of the system. For the exhaustive service discipline, we study the batch sojourn-time, which is defined as the time from an arrival epoch until service completion of the last customer in the batch. We obtain exact expressions for the Laplace–Stieltjes transform of the steady-state batch sojourn-time distribution, which can be used to determine the moments of the batch sojourn-time and, in particular, its mean. However, we also provide an alternative, more efficient way to determine the mean batch sojourn-time, using mean value analysis. We briefly show how our framework can be applied to other service disciplines: locally gated and globally gated. Finally, we compare the batch sojourn-times for different service disciplines in several numerical examples. Our results show that the best performing service discipline, in terms of minimizing the batch sojourn-time, depends on system characteristics

Workloads and waiting times in single-server systems with multiple customer classes

One of the most fundamental properties that single-server multi-class service systems may possess is the property of work conservation. Under certain restrictions, the work conservation property gives rise to a conservation law for mean waiting times, i.e., a linear relation between the mean waiting times of the various classes of customers. This paper is devoted to single-server multi-class service systems in which work conservation is violated in the sense that the server's activities may be interrupted although work is still present. For a large class of such systems with interruptions, a decomposition of the amount of work into two independent components is obtained; one of these components is the amount of work in the corresponding systemwithout interruptions. The work decomposition gives rise to a (pseudo)conservation law for mean waiting times, just as work conservation did for the system without interruptions

Analysis and optimization of vacation and polling models with retrials

We study a vacation-type queueing model, and a single-server multi-queue polling model, with the special feature of retrials. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. Our main focus is on queue length analysis, both at embedded time points (beginnings of glue periods, visit periods and switch- or vacation periods) and at arbitrary time points.Comment: Keywords: vacation queue, polling model, retrials Submitted for review to Performance evaluation journal, as an extended version of 'Vacation and polling models with retrials', by Onno Boxma and Jacques Resin

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