The advantage of relative priority regimes in multi-class multi-server queueing systems with strategic customers
Introduction
It is well known that the -rule is optimal in centrally planned queueing systems. Specifically, suppose that there is a single-server queue that is fed by various types of customers who arrive in accordance with a Poisson process. Let be the mean service time of a class customer and be his waiting cost per unit of time. The -rule says that of all non-preemptive queue regimes, the one that minimizes the mean total (undiscounted) waiting cost is the one that gives absolute priority based on a decreasing order of . This means that upon service completion, the next customer to enter service should be one whose parameter among all other customers present is maximal. It is of course not important which customer among those of this maximal class will be the one to enter. This observation holds also regardless of how many customers of each class are present. See, e.g., [6], p. 125. Suppose there are classes of customers. Then there exists absolute priority orderings of the classes and, as said above, the optimal one among them is the one based on the -rule.
In the above framework the introduction of the option of using lotteries does not change anything. Specifically, suppose that one is not limited to the policies which prioritize the classes, but one is allowed to perform a lottery regarding who should enter next. Yet, under this extended set of policies the optimal policy is still the one based on the rule. This may lead one to conjecture that there is no need for strategies involving lotteries when looking for optimization in queues. However, this conjecture is false. In [4] an example is shown that if customers behave strategically, then a central planner might have such strategy that yields a profit that is strictly higher than any strategy without lotteries. In this example each customer has the option of whether to join the queue or not. Joining is rewarding but it comes with a class-dependent entry fee (in addition to the waiting cost). A profit maximizer who collects the entry fees can assign relative priority parameters to the customers that are based on their class. The next to enter service is selected by a lottery that gives customers entrance probabilities that are proportional to their parameters. As it turns out, the priority parameters affect the joining rate at various classes, which then leads to the corresponding profits. Fine tuning of these parameters leads to an increase in the profits.
This paper shows once again how useful such a relative priority scheme can be in queueing models that are multi-server in addition to being multi-class. To this end, we consider two models. The first is a W-shaped model; see Fig. 1 below. In this model there are two classes of customers and three servers. Each class of customers has its dedicated server and one of the servers can serve both classes.
The second model is an M-shaped model; see Fig. 4 below. In this model there are three classes of customers and two servers. Each server has its exclusive class of customers and one of the classes can be served by both servers. Details are given below.
In a system with centralized planning, the decision makers choose the rules for allocating servers to demand classes so as to achieve the best possible system objectives. In this paper we use these settings to motivate the two models we consider, but in our approach customers select the server to balance their waiting times (if possible) between the servers that are available to them. One may think that giving priority to the class with the highest cost to mean service time ratio (the rule) is optimal, but as we show, this is not necessarily the case. It might be better to give each class only relative priority (in the sense to be defined later). This seeming paradox can be attributed to the fact that customers behave strategically.
Section snippets
Relative priorities, notations, and auxiliary results
In the two models studied in this paper, some of the memoryless servers serve two independent streams of Poisson arrivals of customers. The entrance regime for these servers is that of relative priority. By that we mean that whenever the server is ready to commence servicing the next customer, and there are customers of type in the queue in front of him, , then the next to enter service is the one at the head of the line of type- customers with probability , where
The W model
Consider the following memoryless three-server model. Server- serves at the rate of , . There are two independent streams of Poisson arrivals, stream- with rate , . A fraction of of the customers of stream go to server , . The others (with a total rate of ) go to server . See Fig. 1.
Server uses the first-come first-served entrance policy, . The entrance regime for server 3 is that of relative priority, with priority parameter for customers from
The M model
Consider the following two-server memoryless queueing system. Server works at a service rate of , , and is fed by a Poisson stream of customers, . There is a third Poisson stream of customers with a rate of . A fraction of the third stream goes to server while the rest goes to server . See Fig. 4. We denote by , and by .
The queueing policy at each queue is that of relative priority. The priority parameter of class customers in their queue is , , while
Extensions
The two models discussed here can be generalized in two directions. First, service distribution at each server can be type-dependent. Second, service times need not be exponential. In fact, we can have both these extensions simultaneously. Indeed, for servers that serve a single class of customers, we get by the famous Khintchine–Pollaczek formula that mean waiting times are monotone decreasing and concave functions of their arrival rates. Mean waiting times in servers that serve multiple
Acknowledgments
The majority of this work was done while the first author was a Ph.D. student in the Department of Statistics and the Federmann Center for the Study of Rationality, the Hebrew University of Jerusalem. Research was partially funded by the Programme of Canadian Studies, Co-sponsored by the Government of Canada and Ralph and Roz Halbert of Toronto.
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