We introduce a rate balance principle for general (not necessarily Markovian) stochastic processe... more We introduce a rate balance principle for general (not necessarily Markovian) stochastic processes. Special attention is given to processes with birth-and-death like transitions, for which it is shown that for any state n, the rate of two consecutive transitions from n -1 to n + 1, coincides with the corresponding rate from n + 1 to n -1. We demonstrate how useful this observation is by deriving well-known, as well as new, results for non memoryless queues with state dependent arrival and service processes. We also use the rate balance principle to derive new results for a state dependent queue with batch arrivals, which is a model with non-birth-and-death like transitions.
A cooperative game with transferable utility is said to be homogeneous of degree one if for any i... more A cooperative game with transferable utility is said to be homogeneous of degree one if for any integer m, the value of cloning m times all players at any given coalition, leads to m times the value of the original coalition. We show that this property coupled with subadditivity, guarantees the nonemptyness of the core of the game and of all its subgames, namely, the game is totally balanced. Examples for games stemming from the areas of retailing and of facility location are given.
For the M/G/1 model, we look into a preemptive priority scheme in which the priority level is dec... more For the M/G/1 model, we look into a preemptive priority scheme in which the priority level is decided by a lottery. Such a scheme has no effect on the mean waiting time in the non-preemptive case (in comparison with the First Come First Served (FCFS) regime, for example). This is not the case when priority comes with preemption. We derived the resulting mean waiting time (which is invariant with respect to the lottery performed) and show that it lies between the corresponding means under the FCFS and the Last Come First Served with Preemption Resume (LCFS-PR) (or equivalently, the Egalitarian Processor Sharing (EPS)) schemes. We also derive an expression for the Laplace-Stieltjes transform for the time in the system in this model. Finally, we show how this priority scheme may lead to an improvement in the utilization of the server when customer decide whether or not to join.
European Journal of Operational Research, Sep 1, 2019
We consider the M n /G n /1 queue with vacations and exhaustive service in which the server takes... more We consider the M n /G n /1 queue with vacations and exhaustive service in which the server takes (repeated) vacations whenever it becomes idle, the service time distribution is queue length dependent, and the arrival rate varies both with the queue length and with the status of the server, being busy or on vacation. Using a rate balance principle, we derive recursive formulas for the conditional distribution of residual service or vacation time given the number of the customers in the system and the status of the server. We also derive a closed form expression for the steady-state distribution as a function of the probability of an empty system. As an application of the above, we provide a recursive computation method for the Nash equilibrium joining strategies to the observable M/G/1 queue with vacations.
We derive a revenue-maximizing scheme that charges customers who are homogeneous with respect to ... more We derive a revenue-maximizing scheme that charges customers who are homogeneous with respect to their waiting cost parameter for a random fee in order to become premium customers. This scheme incentivizes all customers to purchase priority, each at his/her drawn price. We also design a revenue-maximizing scheme for the case where customers are heterogeneous with respect to their waiting cost parameter. Now lower cost parameter customers are encouraged to join the premium class at a low price: Given that, those with high cost parameter would be willing to pay even more for this privilege.
We derive the waiting time distribution of the lowest class in an accumulating priority (AP) queu... more We derive the waiting time distribution of the lowest class in an accumulating priority (AP) queue with positive Lévy input. The priority of an infinitesimal customer (particle) is a function of their class and waiting time in the system, and the particles with the highest AP are the next to be processed. To this end we introduce a new method that relies on the construction of a workload overtaking process and solving a first-passage problem using an appropriate stopping time.
We consider a single multi-server memoryless service station. Servers have heterogeneous service ... more We consider a single multi-server memoryless service station. Servers have heterogeneous service rates. Arrivals are routed to one of the servers, and the routing decisions are not based on the queue lengths. We consider two criteria for routing selection: the (Nash) equilibrium, under which each customer minimizes his own mean waiting time, given the behavior of the others; and social optimization, where the routing minimizes the average mean waiting time across all arrivals. The ratio between the social costs of these two routings is called the price of anarchy (PoA). We show that the PoA is upper bounded by the number of servers used in the socially optimal outcome. We also show that this bound is tight.
Multiplicity of solutions is typical to systems where the individual's tendency to act in a certa... more Multiplicity of solutions is typical to systems where the individual's tendency to act in a certain way increases when more of the other individuals in the population act in this way. We provide a detailed analysis of a queueing model in which two priority levels can be purchased. In particular, we compute all of the Nash equilibrium strategies (pure and mixed) of the threshold type.
Consider a population of customers each of which needs to decide independently when to arrive to ... more Consider a population of customers each of which needs to decide independently when to arrive to a facility that provides a service during a fixed period of time, say a day. This is a common scenario in many service systems such as a bank, lunch at a cafeteria, music concert, flight check-in and many others. High demand for service at a specific time leads to congestion that comes at a cost, e.g., for waiting, earliness or tardiness. Queueing Theory provides tools for the analysis of the waiting times and associated costs. If customers have the option of deciding when to join the queue, they will face a decision dilemma of when to arrive. The level of congestion one suffers from depends on others behavior and not only that of the individual under consideration. This fact leads customers to make strategic decisions regarding their time of arrival. In addition, multiple decision makers that affect each other's expected congestion, call for non-cooperative game theoretic analysis of this strategic interaction. This common daily scenario has prompted a research stream pioneered by the ?/M/1 model of Glazer and Hassin [17] that first characterized an arrival process to a queue as a Nash equilibrium solution of a game. This survey provides an overview of the main results and developments in the literature on queueing systems with strategic timing of arrivals. Another issue is that of social optimality, namely the strategy profile used by customers that optimizes their aggregate utility. In particular, we review results concerning the price of anarchy (PoA), which is the ratio between the socially optimal and the equilibrium utilities.
The conventional definition of a cooperative game G(N,V ) with a set of players N = {1, . . . , n... more The conventional definition of a cooperative game G(N,V ) with a set of players N = {1, . . . , n} and a characteristic function V, is quite rigid to alterations of the set of players N . Moreover, it may necessitate a large input of size that is exponential in n. However, the characteristic function of many games allows a simple, efficient and flexible presentation of the game. Here we deal with a set of games that we call regular games, which have a simple presentation: In regular games each player is characterized by a vector of quantitative properties, and the characteristic function value of a coalition depends only on the vectors of properties of its members. We show that some regular games in which players can cooperate with respect to some of their resources and whose immediate formulation does not fit the framework of market games, can nevertheless be transformed into the form of market games and hence they are totally balanced. In particular, they lead to a core allocation...
We consider a memoryless unobservable single-server queue where customers are homogeneous with re... more We consider a memoryless unobservable single-server queue where customers are homogeneous with respect to their reward (due to service completion) and with respect to their cost per unit of time of waiting. Left to themselves, it is well known that in equilibrium they will join the queue at a rate that is higher than it is socially optimal. We show that if customers draw a random preemptive priority parameter prior to deciding whether or not to join, the resulting equilibrium joining rate coincides with the socially optimal one. We also introduce some variations of this regulation scheme and review a few existing schemes from the literature. We suggest a classification of all these schemes, based on a few key properties, and use it to compare our new schemes with the existing ones.
I received much help in composing this solution set from Yoav Kerner, Binyamin Oz and Liron Ravne... more I received much help in composing this solution set from Yoav Kerner, Binyamin Oz and Liron Ravner. Credit is given when due next to the appropriate questions. I am indebted to all three of them.
Consider a population of customers each of which needs to decide independently when to arrive to ... more Consider a population of customers each of which needs to decide independently when to arrive to a facility that provides a service during a fixed period of time, say a day. This is a common scenario in many service systems such as a bank, lunch at a cafeteria, music concert, flight check-in and many others. High demand for service at a specific time leads to congestion that comes at a cost, e.g., for waiting, earliness or tardiness. Queueing Theory provides tools for the analysis of the waiting times and associated costs. If customers have the option of deciding when to join the queue, they will face a decision dilemma of when to arrive. The level of congestion one suffers from depends on others behavior and not only that of the individual under consideration. This fact leads customers to make strategic decisions regarding their time of arrival. In addition, multiple decision makers that affect each other's expected congestion, call for non-cooperative game theoretic analysis of this strategic interaction. This common daily scenario has prompted a research stream pioneered by the ?/M/1 model of Glazer and Hassin [17] that first characterized an arrival process to a queue as a Nash equilibrium solution of a game. This survey provides an overview of the main results and developments in the literature on queueing systems with strategic timing of arrivals. Another issue is that of social optimality, namely the strategy profile used by customers that optimizes their aggregate utility. In particular, we review results concerning the price of anarchy (PoA), which is the ratio between the socially optimal and the equilibrium utilities.
We derive a revenue-maximizing scheme that charges customers who are homogeneous with respect to ... more We derive a revenue-maximizing scheme that charges customers who are homogeneous with respect to their waiting cost parameter for a random fee in order to become premium customers. This scheme incentivizes all customers to purchase priority, each at his/her drawn price. We also design a revenue-maximizing scheme for the case where customers are heterogeneous with respect to their waiting cost parameter. Now lower cost parameter customers are encouraged to join the premium class at a low price: Given that, those with high cost parameter would be willing to pay even more for this privilege.
We consider the M n /G n /1 queue with vacations and exhaustive service in which the server takes... more We consider the M n /G n /1 queue with vacations and exhaustive service in which the server takes (repeated) vacations whenever it becomes idle, the service time distribution is queue length dependent, and the arrival rate varies both with the queue length and with the status of the server, being busy or on vacation. Using a rate balance principle, we derive recursive formulas for the conditional distribution of residual service or vacation time given the number of the customers in the system and the status of the server. We also derive a closed form expression for the steady-state distribution as a function of the probability of an empty system. As an application of the above, we provide a recursive computation method for the Nash equilibrium joining strategies to the observable M/G/1 queue with vacations.
We consider an unobservable M/M/1 queue where customers are homogeneous with respect to service v... more We consider an unobservable M/M/1 queue where customers are homogeneous with respect to service valuation and cost per unit time of waiting. It is well known that left to themselves, in equilibrium, customers join the queue at a rate higher than is socially optimal. Hence, regulation schemes, under which the resulting equilibrium joining rate coincides with the socially optimal one, should be considered. We suggest a classification of regulation schemes based on a few desired properties and use it to classify schemes from the existing literature. To the best of our knowledge, none of the existing schemes possesses all of the properties, and in this paper we suggest such a scheme. Its novelty is in assigning random priorities to customers, prior to their decision whether to join or balk. We also introduce variations of this regulation scheme as well as others that are also based on randomization. The e-companion is available at https://doi.org/10.1287/mnsc.2017.2728 . This paper was ...
Queues in which customers who belong to different classes, have different priority levels are an ... more Queues in which customers who belong to different classes, have different priority levels are an old subject. Usually one looks for the performance of each class given its priority level. We suggest here a new model. Specifically, we consider the M/G/1 queue model in which all customers are identical ex-ante but prior to joining the queue, they draw a random (preemptive) priority level. We derive the Laplace-Stieltjes transform (LST) of a customer given his drawn priority parameter. From that the LST of an arbitrary customer can be integrated out. We present a a number of proofs which give some insight to the model. Special attention in given to the case of exponential service (the M/M/1 queue) and to finding the first moment of waiting. In particular, we show that the model is 'a middle of the road one' in the sense that the mean sojourn time lies between the corresponding means under the FCFS and the Last come first served (LCFS-PR) (or equivalently, the Egalitarian Processor Sharing (EPS)) schemes. Finally, we show how the new scheme may lead to an improvement in the utilization of the server when customer decide whether or not to join. We conclude with a few words on the corresponding model but without preemption.
was the first to observe that in a single-server memoryless queue, customers who inspect the queu... more was the first to observe that in a single-server memoryless queue, customers who inspect the queue length upon arrival and accordingly decide whether to join or not may join even if from the social point of view they are worse off. The question then is how to mechanically design the system such that customers will join only queue lengths that are advised by society, while still minding their own selfish utility. After reviewing some existing mechanisms (some involving money transfers and some not), we suggest novel ones that do not involve money transfers. They possess some advantages over the existing ones, which we itemize.
Proceedings of the 8th International Conference on Performance Evaluation Methodologies and Tools, 2015
For the M/G/1 model, we look into a preemptive priority scheme in which the priority level is dec... more For the M/G/1 model, we look into a preemptive priority scheme in which the priority level is decided by a lottery. Such a scheme has no effect on the mean waiting time in the non-preemptive case (in comparison with the First Come First Served (FCFS) regime, for example). This is not the case when priority comes with preemption. We derived the resulting mean waiting time (which is invariant with respect to the lottery performed) and show that it lies between the corresponding means under the FCFS and the Last Come First Served with Preemption Resume (LCFS-PR) (or equivalently, the Egalitarian Processor Sharing (EPS)) schemes. We also derive an expression for the Laplace-Stieltjes transform for the time in the system in this model. Finally, we show how this priority scheme may lead to an improvement in the utilization of the server when customer decide whether or not to join.
Uploads
Papers by Moshe Haviv