The Concert Queueing Game with a Random Volume of Arrivals Sandeep Juneja Tata Institute of Fundamental Research Mumbai juneja@tifr.res.in Tushar Raheja Indian Institute of Technology Delhi tushar@raheja.org Nahum Shimkin Technion - Israel Institute of Technology Haifa shimkin@ee.technion.ac.il Abstract—We consider the concert queueing game in the ﬂuid framework, where the service facility opens at a speciﬁed time, the customers are particles in a ﬂuid with homogeneous costs that are linear and additive in the waiting time and in the time to service completion, and wish to choose their own arrival times so as to minimize their cost. This problem has recently been analyzed under the assumption that the total volume of arriving customers is deterministic and known beforehand. We consider here the more plausible setting where this volume may be random, and only its probability distribution is known beforehand. In this setting, we identify the unique symmetric Nash equilibrium and show that under it the customer behavior signiﬁcantly differs from the case where such uncertainties do not exist. While, in the latter case, the equilibrium proﬁle is uniform, in the former case it is uniform up to a point and then it tapers off. We also solve the associated optimization problem to determine the socially optimal solution when the central planner is unaware of the actual amount of arrivals. Interestingly, the Price of Anarchy (ratio of the social cost of the equilibrium solution to that of the optimal one) for this model turns out to be two exactly, as in the deterministic case, despite the different form of the social and equilibrium arrival proﬁles. I. I NTRODUCTION Customers going to a rock concert or a movie theater need to resolve the following dilemma: Going early involves encoun- tering a rush to get the best seats, going late involves sacriﬁce in the viewing experience. Evening commuters often face the trade-off between reaching home late from work or getting caught in the evening rush hour. Similar trade-offs govern queueing behavior in a busy cafeteria: People may prefer to eat as soon as the cafeteria opens at lunch time or they may choose to stay hungry and eat later when the waiting is less but the food quality may deteriorate. We refer to this ‘queue arrival timing problem’ as the concert queueing problem (see [10] and [9]). This problem is especially important when the number of potential customers involved is large. Typically, even when the size of population coming to a queue is large, it may still be substantially variable. In this paper we consider this concert queueing problem in the ﬂuid framework. Here each customer is a particle or a point in a continuum that needs to decide when to arrive to a queue where the server opens service at a speciﬁed time. The arrivals are non-cooperative, their cost structure is homogeneous and is linear and additive in the waiting time and in the time to service. The customers can arrive before or after the server opens for service, and are served in a ﬁrst come ﬁrst serve manner. This problem was recently considered in [10], where the total volume of customers is assumed to be ﬁxed and known beforehand to arriving customers. This ﬂuid model approximates the actual scenario where the total number of customers is ﬁnite but large and more or less constant (see [11] for a proof of convergence of the equilibrium proﬁle in the discrete queueing model to that of the associated ﬂuid model as the number of customers increases to inﬁnity). This basic ﬂuid model has been extended to multiple classes of customers [9], parallel and serial queues [8], and different opening and closing conditions [7]. In this paper, we analyze a more realistic scenario in the ﬂuid setting, where the volume of arriving customers may be random, and only its probability distribution is known upfront. In [10] and [9], the authors show that there exists a unique Nash equilibrium arrival proﬁle that corresponds to customers arriving uniformly over a speciﬁed interval. They further show that the price of anarchy (the ratio of the social cost of the worst Nash equilibrium to the optimal one) in their framework equals 2. As mentioned above, we extend this framework to allow for random arrival volume. Under this extension, we derive the unique symmetric equilibrium proﬁle for customer arrival instances. We note that this differs signiﬁcantly from the arrival proﬁle when volume of arrivals is ﬁxed. Speciﬁ- cally, we show that in the random setting, the unique Nash equilibrium proﬁle is uniform only up to a point and then it tapers off as a function of time. Thus, customers have a higher arrival density in the beginning of the arrival period than at its end. We also explicitly evaluate the cost incurred by each customer in equilibrium, and verify that uncertainly in the arrival volume tends to increase this cost. We also consider the problem of determining the socially optimal solution in this setting when the central planner is unaware of the volume of the arriving trafﬁc, but can dictate the distribution of arrival times for those who do arrive. This problem may be of independent interest in various settings. For instance, when a central planner gives appointments to arriving customers and a random amount of customers show up. It is also useful in ascertaining the level of inefﬁciency of the equilibrium proﬁle through the computation of PoA. We note that unlike in the case where the arrival volume is ﬁxed, when it is allowed to be random, the social optimal solution may VALUETOOLS 2012, October 09-12, Cargèse, France Copyright © 2012 ICST DOI 10.4108/valuetools.2012.250166