Performance analysis of polling systems with retrials and glue periods

We consider gated polling systems with two special features: (i) retrials, and (ii) glue or reservation periods. When a type-$i$ customer arrives, or retries, during a glue period of station $i$, it will be served in the next visit period of the server to that station. Customers arriving at station $i$ in any other period join the orbit of that station and retry after an exponentially distributed time. Such polling systems can be used to study the performance of certain switches in optical communication systems. For the case of exponentially distributed glue periods, we present an algorithm to obtain the moments of the number of customers in each station. For generally distributed glue periods, we consider the distribution of the total workload in the system, using it to derive a pseudo conservation law which in its turn is used to obtain accurate approximations of the individual mean waiting times. We also consider the problem of choosing the lengths of the glue periods, under a constraint on the total glue period per cycle, so as to minimize a weighted sum of the mean waiting times

Iterative approximation of k-limited polling systems

The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) k-limited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logistics. Knowledge of the queue length distributions is needed to operate the system properly. The multi-queue polling system is decomposed into single-queue vacation systems with k-limited service and state-dependent vacations, for which the vacation distributions are computed in an iterative approximate manner. These vacation models are analyzed via matrix-analytic techniques. The accuracy of the approximation scheme is verified by means of an extensive simulation study. The developed approximation turns out be accurate, robust and computationally efficient

Polling, production & priorities

The present monograph focuses on the so-called stochastic economic lot scheduling problem(SELSP), which deals with the make-to-stock production of multiple standardizedproducts on a single machine with limited capacity under random demands, possibly randomsetup times and possibly random production times. In the SELSP, a production policyis needed which describes for each possible state of the system whether to continue productionof the current product, whether to switch to another product or whether to idlethe machine. The objective of the present monograph is the development and the analysisof mathematical models that capture the behavior of the class of fixed-sequence base-stockpolicies. For given base-stock levels, it is shown that the analysis of a fixed-sequence basestockpolicy is tantamount to the analysis of the queue length distribution in a classicalqueueing model, the so-called polling system.The focus of the current research is mainly on the lot-sizing decision: what should thelength of the production run be? Within the context of this lot-sizing decision the presentmonograph is, in particular, concerned with the evaluation and comparison of the traditionalexhaustive and gated lot-sizing policies, on the one hand, and the more sophisticatedquantity-limited lot-sizing policy, on the other hand. The latter offers the possibility toprioritize among the different products for improving total system performance throughbounding the lengths of the production runs. Evaluation and optimization of these lotsizingdisciplines are achieved through state-of-the-art analysis of several polling systems.We study two research objectives as summarized below.Research objective 1. Development of a unifying exact framework for the analysis ofthe exhaustive and gated lot-sizing policies in terms of the average work-in-progress (WIP)levels under the assumption of Poisson demand processes. ¤In Chapter 3 an exact Mean Value Analysis (MVA) framework for the exhaustive andgated lot-sizing disciplines is presented, which computes the average WIP levels by exploitingdirect mean value arguments. Within this framework the individual WIP levels canbe efficiently obtained via the solution of a sparse set of linear equations, whereas for thetotal WIP level a closed-form expression is presented.The MVA framework gives rise to explicit closed-form expressions, allowing for back-ofthe-envelope calculations, for the individual WIP levels in the asymptotic regime of highutilization of capacity due to either customer demands or setup times. These expressionsexplicitly show the impact of all input parameters, yield insensitivity and monotonicityproperties and unearth the (dis)similarities between the two sources of high utilization.In particular, it is shown that the exhaustive and gated lot-sizing disciplines display undesirablebehavior if the utilization rate is high due to customer demand, which revealsitself, for example, in difficulties in the coordination between stages within the productionprocess.Motivated by the practical significance of the large setup times regime, we study thisregime in more detail for a general class of branching-type lot-sizing policies by usingmore advanced techniques. The most remarkable result of this analysis is the fact thatthe stochastic system converges to its deterministic counterpart in the limit of increasingsetup times implying that the exhaustive lot-sizing policy is optimal in terms of the WIPlevels and that, thus, production runs should not be bounded in systems with extremelylarge setup times. For general settings, the latter conclusion does not always hold whichwe analytically show in the analysis of the second research objective.Research objective 2. Development of an efficient and accurate approximate tool forthe analysis of the quantity-limited lot-sizing policy under the assumption of general demandprocesses. ¤In order to gain insights into the impact of bounding production runs and not to be divertedby other effects, Chapter 4 starts the analysis with a basic occurrence of the SELSPin an exact way. That is, we analyze a two-product system, in which a high-priority productis produced exhaustively and a low-priority product according to the quantity-limitedservice strategy. In this model, we observe significant cost reductions by application ofthe quantity-limited policy, compared to the standard exhaustive policies, indicating thepotential of the quantity-limited service discipline as lot-sizing rule in production environments.The results obtained in the two-product case provide us with theoretical evidence thatthe quantity-limited strategy may lead to considerable cost reductions compared to thewidely used (standard) exhaustive policy. Therefore, in Chapter 4 we develop an efficientand accurate approximate decomposition approach for the evaluation of quantity-limitedlot-sizing policies under the most general imaginable assumptions, i.e., general number ofproducts each with their own quantity limit in an environment with generally distributedarrival, service time and setup time distributions. The accuracy of the approximationscheme is verified by means of an extensive simulation study.The last part of Chapter 4 is devoted to a numerical simulation study assessing thequality of the quantity-limited lot-sizing policy as tool for prioritizing among products. Itis shown that the quantity-limited lot-sizing policy outperforms the standard exhaustivepolicy leading to improvements in system performance for a variety of environments.Finally, we would like to emphasize that the results of the present monograph are certainlynot limited to the described production setting, but may be used in the design andoptimization phase of many other fields of application such as communication, maintenance,manufacturing and transportation

Editorial polling systems

From the issue entitled Polling Stations

Critically loaded k-limited polling systems

We consider a two-queue polling model with switch-over times and k-limited service (serve at most ki customers during one visit period to queue i) in each queue. The major benefit of the k-limited service discipline is that it - besides bounding the cycle time - effectuates prioritization by assigning different service limits to different queues. System performance is studied in the heavy-traffic regime, in which one of the queues becomes critically loaded with the other queue remaining stable. By using a singular-perturbation technique, we rigorously prove heavy-traffic limits for the joint queue-length distribution. Moreover, it is observed that an interchange exists among the first two moments in service and switch-over times such that the HT limits remain unchanged. Not only do the rigorously proven results readily carry over to N( ≥ 2) queue polling systems, but one can also easily relax the distributional assumptions. The results and insights of this note prove their worth in the performance analysis of Wireless Personal Area Networks (WPAN) and mobile networks, where different users compete for access to the shared scarce resources

Case study of a batch-production/inventory system

The plant of BASF under consideration consists of multiple parallel production lines, which produce multiple products in a make-to-stock fashion for process industry. Complicating fac- tors for planning are the stochastic demand, setup times, batch processing and finite buffer capacities. The main contribution is the development of a three-stage methodology integrating production and inventory decisions, which can be used for the evaluation and optimization of a wide range of batch-production/inventory systems. Implementation of this methodology in a decision support tool enabled us to identify major opportunities for improvement of current practice

Case study of a batch-production and inventory system

No abstract