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We study a closed queueing system that offers two types of service. After obtaining service the customer is active for a period of time whose random duration depends on the service type: the longer service type enables longer activity time. Users can choose a service type every time they are in service, and we are interested in the system efficiency which is the average number of active users. Applications include but are not limited to inventory control, electrical vehicle charging, mobile robotics charging, and machine maintenance. We prove that choosing the more efficient service is not always optimal. We demonstrate that if we constrain the strategy to be a threshold strategy depending on the number of inactive users, then the resulting maximal efficiency is not significantly lower than the maximal efficiency. We also study individuals' optimal selection of the service type, and observe that individuals and the manager have opposite incentives: when the queue is long, individuals choose the long service, while the manager prefers the short service in this case., Comment: 41 pages, 11 figures

A consumer who wants to consume a good in a particular period may nevertheless attempt to buy it earlier if he is concerned that in delaying he would find the good already sold. This paper considers a model in which the good may be offered in two periods; the period in which all consumers most value the good (period 2), and an earlier period (period 1). Examining the profit-maximizing strategy of the firm under unbounded demand, we find that even with no cost of making the product available early, the firm does not profit, and usually loses, by making the product available early. Interestingly, the price that maximizes profits induces all arrivals to occur early, or all arrivals to occur late, depending on the parameters. The firm would not set a price which induces consumers to arrive in both periods. In particular, if the firm controls the penalty for arriving early, then it should set a high penalty so that no one arrives early. The Nash equilibrium behavior of consumers, when deciding if and when to arrive is more complicated than one may suppose, and can generate some unexpected behavior. For example, when there is unbounded demand, most potential consumers decide not to arrive at all. Additionally, the arrival rate may decline with the surplus a person gets from buying the good. Surprisingly, we find that an increase in the number of units for sale increases the number of consumers who arrive early. Moreover, we find that the profit-maximizing price increases with the number of units offered for sale. This too is unexpected as an increase in supply often results in price reduction. In our case, an increase in the number of units on sale also increases demand, and the seller may profit by increasing the price. In the single-unit case, we give closed solutions for the equilibrium customer behavior and profit-maximizing firm strategy and conduct sensitivity analysis.

We consider a game of decentralized timing of jobs to a single server (machine) with a penalty for deviation from a due date, and no delay costs. The jobs' sizes are homogeneous and deterministic. Each job belongs to a single decision maker, a customer, who aims to arrive at a time that minimizes his deviation penalty. If multiple customers arrive at the same time then their order of service is determined by a uniform random draw. We show that if the cost function has a weighted absolute deviation form then any Nash equilibrium is pure and symmetric, that is, all customers arrive together. Furthermore, we show that there exist multiple, in fact a continuum, of equilibrium arrival times, and provide necessary and sufficient conditions for the socially optimal arrival time to be an equilibrium. The base model is solved explicitly, but the prevalence of a pure symmetric equilibrium is shown to be robust to several relaxations of the assumptions: restricted server availability, inclusion of small waiting costs, stochastic job sizes, randomly sized population, heterogeneous due dates, and non-linear deviation penalties.

This work presents a variation of Naor's strategic observable model (1969), by adding a component of customer heterogeneity induced by the location of customers in relation to the server. Accordingly, customers incur a travel cost which depends linearly on the distance of the customer from the server. The arrival of customers with distances less than x is assumed to be a Poisson process with rate lambda(x)=int_0^x h(y)dy<\infty, where h(y) is a nonnegative intensity function of the distance y. In a loss system M/G/1/1 we define the threshold Nash equilibrium strategy x_e and the optimal social threshold strategy x^*. We show that if the rate of arriving customers is bounded then PoA converges to 1 when x_e \to\infty, i.e., in the limit there is no difference between the social and equilibrium optimal benefits. The rest of the paper is dedicated for the case in which the rate of arriving customers is unbounded. We develop an explicit formula to calculate lim_{x_e\to \infty}PoA when it exists. We present sufficient conditions for the limit to exist and for the existence of a simple formula for calculating it. We prove that if the relation between two intensity functions converges to a positive constant, then the corresponding limits of PoA coincide. If, on the other hand, one intensity function is larger than the other from some point on, then under certain conditions the limit of PoA (if exists) will be larger for the larger intensity function. For all intensity functions h, we prove that if h converges to a constant then PoA converges to 2, and that if from some point on h decreases (increases) monotonically then the limit of PoA, if exists, is smaller (larger) than 2. In a system with a queue we prove that the price of anarchy may be unbounded already in the simple case of uniform arrival, namely h~c, where c>0.

We consider a service system with an infinite number of exponential servers sharing a finite service capacity. The servers are ordered according to their speed, and arriving customers join the fastest idle server. A capacity allocation is an infinite sequence of service rates. We study the probabilistic properties of this system by considering overflows from sub-systems with a finite number of servers. Several stability measures are suggested and analysed. The tail of the series of service rates that minimizes the average expected delay (service time) is shown to be approximately geometrically decreasing. We use this property in order to approximate the optimal allocation of service rates by constructing an appropriate dynamic program.

Let G be an edge weighted undirected graph. For every pair of nodes consider the shortest cycle containing these nodes in G. The cycle diameter of G is the maximum length of a cycle in this set. Let H be a directed graph obtained by directing the edges of G. The cycle diameter of H is similarly defined except for that cycles are replaced by directed closed walks. Is there always an orientation H of G whose cycle diameter is bounded by a constant times the cycle diameter of G? We prove this property for planar graphs. These results have implications on the problem of approximating an orientation with minimum diameter