arXiv subjects

Igor Kleiner

Publications and source records attributed to Igor Kleiner.

4 recordsLinked to original sources

arXiv2026

One Empty Locker and Two Inspections: An Exact Optimal Team Strategy

We study a cooperative search game with $N$ lockers, $N-1$ labelled objects, one empty locker, and $N-1$ players. Player $i$ seeks object $i$ and may inspect at most two lockers; the second inspection may depend on the content of the first. The players may coordinate beforehand but receive no information about the searches of other players after play begins. We prove that the maximum probability that every player finds the assigned object is $I_N/N!$, where $I_N$ is the number of involutions of $N$ elements. An optimal strategy represents the blank by the common fictitious symbol $N$ and follows pointers: player $i$ first opens locker $i$, then opens the locker whose number was observed. The upper bound holds for every deterministic or randomized adaptive strategy and follows from a deletion lemma and two recurrences. We give explicit examples, identify the three-door case with an isomorphic "Return of Monty Hall" game, and report a reproducible mixed-integer verification for $N=2,3,4,5$. The computation is independent of, and not needed for, the proof.

math.CO↗

arXiv2026

An Exact Solution of the Two-Ball Multi-Look Search Game with Three Boxes and Heterogeneous Costs

A Hider distributes two identical balls among three boxes whose search costs satisfy $a\ge b\ge c>0$. A Searcher opens boxes adaptively until both balls are found; every opening incurs the corresponding box cost and recovers at most one ball. We give an exact solution of this heterogeneous multi-look search-cost game. The value is the maximum of three explicit rational functions. In the three parameter regimes, an optimal Hider strategy is the product-form distribution restricted respectively to three, five, or all six placements. Every non-wasteful deterministic Searcher policy is payoff-equivalent to one of $72$ elementary decision trees, which induce only $42$ distinct opening-count profiles. An analytic argument settles the first regime. Two exact Searcher mixtures settle the second, and four settle the third. Feasibility of the parameterized mixtures over the full cost region is established by exact rational Bernstein-basis certificates. Independent implementations reproduce the policy set, the zero-sum linear-program values, and all $154$ dyadic nodes examined by the Bernstein verifier.

math.OC↗

arXiv2021

A Decomposition Property for an $M^{X}/G/1$ Queue with Vacations

We introduce a queueing system that alternates between two modes, so-called {\it working mode} and {\it vacation mode}. During the working mode the system runs as an $M^{X}/G/1$ queue. Once the number of customers in the working mode drops to zero the vacation mode begins. %Then working system becomes empty the vacation phase begins. During the vacation mode the system runs as a general queueing system (a service might be included) which is different from the one in the working mode. The vacation period ends in accordance with a given stopping rule, and then a random number of customers are transferred to the working mode. For this model we show that the conditional probability generating function of the number of customers given that the system is in the working mode is a product of three terms. This decomposition result puts under the same umbrella some models that have already been introduced in the past as well as some new models.

math.PR↗

arXiv2018

FCFS Parallel Service Systems and Matching Models

We consider three parallel service models in which customers of several types are served by several types of servers subject to a bipartite compatibility graph, and the service policy is first come first served. Two of the models have a fixed set of servers. The first is a queueing model in which arriving customers are assigned to the longest idling compatible server if available, or else queue up in a single queue, and servers that become available pick the longest waiting compatible customer, as studied by Adan and Weiss, 2014. The second is a redundancy service model where arriving customers split into copies that queue up at all the compatible servers, and are served in each queue on FCFS basis, and leave the system when the first copy completes service, as studied by Gardner et al., 2016. The third model is a matching queueing model with a random stream of arriving servers. Arriving customers queue in a single queue and arriving servers match with the first compatible customer and leave immediately with the customer, or they leave without a customer. The last model is relevant to organ transplants, to housing assignments, to adoptions and many other situations. We study the relations between these models, and show that they are closely related to the FCFS infinite bipartite matching model, in which two infinite sequences of customers and servers of several types are matched FCFS according to a bipartite compatibility graph, as studied by Adan et al., 2017. We also introduce a directed bipartite matching model in which we embed the queueing systems. This leads to a generalization of Burke's theorem to parallel service systems.

math.PR↗