Assigning multiple job types to parallel specialized servers

In this paper methods of mixing decision rules are investigated and applied to the so-called multiple job type assignment problem with specialized servers. This problem is modeled as continuous time Markov decision process. For this assignment problem performance optimization is in general considered to be difficult. Moreover, for optimal dynamic Markov decision policies the corresponding decision rules have in general a complicated structure not facilitating a smooth implementation. On the other hand optimization over the subclass of so-called static policies is known to be tractable. In the current paper a suitable static decision rule is mixed with dynamic decision rules which are selected such that these rules are relatively easy to describe and implement. Some mixing methods are discussed and optimization is performed over corresponding classes of so-called mixing policies. These mixing policies maintain the property that they are easy to describe and implement compared to overall optimal dynamic Markov decision policies. Besides for all investigated instances the optimized mixing policies perform substantially better than optimal static policies

Optimal mixing of Markov decision rules for MDP control

In this article we study Markov decision process (MDP) problems with the restriction that at decision epochs, only a finite number of given Markov decision rules are admissible. For example, the set of admissible Markov decision rules D could consist of some easy-implementable decision rules. Additionally, many open-loop control problems can be modeled as an MDP with such a restriction on the admissible decision rules. Within the class of available policies, optimal policies are generally nonstationary and it is difficult to prove that some policy is optimal. We give an example with two admissible decision rules - D={

Note on the convexity of the stationary waiting time as a function of the density

It is proved that the total average waiting time is a convex function of the routing densities for regular routing to parallel queues

Orders and bounds for response times

Rapport interne.Which of two deterministic periodic admission sequences (periodic sequences of nonnegative integers) gives the smaller average expected waiting time has already been investigated. A partial order, called the cone order has been introduced, and it is shown that the average waiting time and more generally any multimodular function is monotone with respect to the cone order. It is natural to define a multimodular order by requiring that any multimodular function is monotone. In contrast to stochastic cases, where the convex order for stochastic admission sequences is used, we consider deterministic admission sequences in this paper. Note that deterministic admission sequences can only be ordered for all multimodular functions if they are equal. Therefore we consider multimodular functions with a fixed minimal point as we did with the cone order. We introduce the multimodular order and we show that the cone order is equivalent to the multimodular order. In section. the shift invariant counterparts of these orders are studied and it is shown that the regular admission sequence is the minimal point. For the optimal routing problem to n queues we derive that a lower bound is obtained by using regular admission sequences (or what is the same bracket sequences) with 'minimizing' routing fractions (densities) as admission sequences to all queues. But generally only in the routing to $n=2$ queues, the routing fractions will be balanceable and only in that case the admission sequences can be glued together and be made to a feasible routing policy

The Unique-lowest Sealed-bid Auction

Unique-lowest sealed-bid auctions are auctions in which participation is endogenous and the winning bid is the lowest bid among all unique bids. Such auctions admit very many Nash equilibria (NEs) in pure and mixed strategies. The two-bidders' auction is similar to the Hawk-Dove game, which motivates to study symmetric NEs: Properties and comparative statics are derived and the symmetric NE with the lowest expected gains is the maximin in symmetric strategies, which allows computation through a mathematical program. Simulations provide numerical evidence that the symmetric NE with the lowest expected gains is the unique limit point of the replicator dynamics.Auctions; Sealed-Bid Auction; Evolutionary Stability; Endogenous Entry; Maximin

Approximate Results for a Generalized Secretary Problem

This discussion paper resulted in a publication in 'Probability in the Engineering and Informational Sciences' , 25(2), 157-69. A version of the classical secretary problem is studied, in which one is interested in selecting one of the b best out of a group of n differently ranked persons who are presented one by one in a random order. It is assumed that b is bigger than or equal to 1 is a preassigned number. It is known, already for a long time, that for the optimal policy one needs to compute b position thresholds, for instance via backwards induction. In this paper we study approximate policies, that use just a single or a double position threshold, albeit in conjunction with a level rank. We give exact and asymptotic (as n goes to infinity) results, which show that the double-level policy is an extremely accurate approximation.Secretary Problem, Dynamic Programming, Approximate Policies