We consider a memoryless single station service system with servers S = {m 1 , . . . , m K }, and with job types C = {a, b, . . .}. Service is skill-based, so that server m i can serve a subset of job types C(m i ). Waiting jobs are served on a first-come-firstserved basis, while arriving jobs that find several idle servers are assigned to a feasible server randomly. We show that there exist assignment probabilities under which the system has a product-form stationary distribution, and obtain explicit expressions for it. We also derive waiting time distributions in steady state.

We consider an infinite sequence of items of types C = {c1, . . . , cI }, and another infinite sequence of items of types S = {s1, . . . , sJ }, and a bipartite graph G of allowable matches between the types. Matching the two sequences on a first come first served basis defines a unique infinite matching between the sequences. For (ci, sj) ∈ G we define the matching rate rc i ,s j as the long term fraction of (ci, sj) matches in the infinite matching, if it exists. We assume that the types of items in the two sequences are i.i.d. with given probability vectors α, β. We describe this system by a Markov chain, obtain conditions for ergodicity, and derive its stationary distribution which is of product form. We show that if the chain is ergodic, then the matching rates exist almost surely, and give a closed form formula to calculate them.

Abstract. In this paper we study a system consisting of two parallel servers with different service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. On arrival a job joins the shortest queue and in case both queues have equal lengths, he joins the first queue with probability q and the second one with probability 1-q, where q is an arbitrary number between 0 and 1. In a previous paper we showed for the symmetric problem, that is for equal service rates and q =1,1, that the equilibrium distribution of the lengths of the two queues can be exactly represented by an infinite sum of product form solutions by using an elementary compensation procedure. The main purpose of the present paper is to prove a similar product form result for the asymmetric problem by using a generalization of the compensation procedure. Furthermore, it is shown that the product form representation leads to a numerically efficient algorithm. Essentially, the method exploits the convergence properties of the series of product forms. Because of the fast convergence an efficient method is obtained with upper and lower bounds for the exact solution. For states further away from the origin the convergence is faster. This aspect is also exploited in the paper.

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This paper develops a two-stage planning procedure for master planning of elective and emergency patients while allocating at best the available hospital resources. Four types of resources are considered: operating theatre, beds in the medium and in the intensive care units, and nursing hours in the intensive care unit. A tactical plan is obtained by minimizing the deviations of the resources consumption to the target levels of resources utilization. Some capacity is reserved for emergency care. To deal with the deviation between actually arriving patients and the average number of patients on which the tactical plan is based, we consider the option of planning a higher number of patients (overplanning). To adapt the tactical plan to the actual stream of elective patients, we also consider flexibility rules. Overplanning and flexibility leads to a weekly schedule of elective patients. This schedule is modified to account for emergency patients. Scheduled elective patients may be cancelled and emergency patients may be sent to other hospitals. Cancellations rules for both types of patients rely on the possibility to exceed the available capacities. Several performance indicators are defined to assess patient service/dissatisfaction and hospital efficiency. Simulation results show a trade-off between hospital efficiency and patient service. We also obtain a rank of the different strategies: overplanning, flexibility and cancellation rules.Keywords: operation theatre planning, intensive and medium care, emergency and elective patients, overplanning, cancellation rules, operational schedule of patients, master surgical schedule.