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This paper analyzes a communication network, used by customers with heterogeneous service requirements. We investigate priority queueing as a way to establish service differentiation. It is assumed that there is an infinite population of customers, who join the network as long as their utility (which is a function of the queueing delay) is larger than the price of the service. We focus on the specific situation with two types of users: one type is delay-sensitive (‘voice’), whereas the other is delay-tolerant (‘data’); these preferences are reflected in their utility curves. Two models are considered: in the first the network determines the priority class of the users, whereas the second model leaves this choice to the users. For both models we determine the prices that maximize the provider’s profit. Importantly, these situations do not coincide. Our analysis uses elements from queueing theory, but also from microeconomics and game theory (e.g., the concept of a Nash equilibrium). We conclude the paper by considering a model in which throughput (rather than delay) is the main performance measure. Again the pricing strategy exploits the heterogeneity in service requirements and willingness-to-pay.

Current usage of data-networks, such as the Internet, is still dominated by ‘traditional’ data services: web browsing, file transfer, remote terminal, electronic mail, etc. These applications do not impose severe requirements on the network, in that they tolerate relatively large packet delays. New Internet applications, e.g., real-time applications such as interactive voice and video, can be characterized as delay-sensitive, and are consequently considerably more demanding. This heterogeneity of the service requirements makes it necessary that the delay-tolerant and delay-sensitive users are handled differently––otherwise all traffic must be handled according to the requirements of the most demanding class, i.e., the real-time class, which will inevitably lead to a network running at a relatively poor utilization level. A possible solution is to give priority to the delay-sensitive traffic in the queues of the network. Shenker [13] further motivates this prioritization and related design issues for the Internet. Pricing. Without an appropriate pricing scheme, any prioritization is useless; if there were no price difference between the priority classes, all users would opt for the high-priority class. In other words: the prices of the priority classes should give users an incentive to join the ‘right’ priority class. In terms of the delay-tolerant user (or, shortly, the data user) and the delay-sensitive user (or, shortly, the voice user): voice users are encouraged to use the high-priority class, whereas data users are given an incentive to join the low-priority class. This is done by imposing a higher charge on the high-priority class. A next question is: how should the network provider choose the prices for both classes in order to maximize its profit? Here two models can be distinguished. In the first model the provider assigns a priority class to each user type––for instance, the provider can decide that the voice customers are directed to the high-priority queue, and the data users to the low-priority queue. This model of ‘dedicated classes’ (or ‘implicit supply of service’, in Shenker’s [13] terminology) is relatively simple to analyze, as the network users have only two alternatives: joining the network or not. The harder, but perhaps more realistic, model is the model with ‘open classes’ (or ‘explicit supply of service’, as it is called in [13]), in which the users can choose between the priority classes. It is not clear beforehand whether the prices that optimize the profit in the dedicated-classes model, are also profit optimizing for the open-classes model. The reason is that the prices found in the dedicated-classes model might lead to a situation in which data (voice) users might appreciate the high-(low-)priority class more. In other words: it is not a priori clear whether the optimal prices from the dedicated-classes model lead to an incentive-compatible situation in the open-classes model. Incentive-compatibility. In economic terms, in the model with open classes, the users of the network are agents, who individually choose between the three alternatives offered, that is, joining the high-priority class, joining the low-priority class, or not using the network at all. The situation in which no user has any incentive to unilaterally change his policy is called a Nash equilibrium [14]. It is not obvious that by making high-priority transfer more expensive than low-priority transfer the voice customers will use the high-priority class and the data customers will use the low priority class; this strongly depends on the price difference between the queues, and the delay performance of both queues. This statement can be made more precise as follows. Let for both types of traffic the mean delay determine the utility experienced by the users. Now the utility curves for data and voice are denoted by ud(·) and uv(·), respectively, and are decreasing in their argument, i.e., the mean delay. Clearly, this mean delay is affected by the number of customers of both types who join both service classes. Suppose that data (voice) customers are assigned to the low-(high-)priority class, leading to mean delays View the MathML source and View the MathML source, respectively. Assume that customers are ‘infinitely divisible’, i.e., we do not restrict ourselves to integer numbers of customers. Then we have a Nash equilibrium if equation(1) View the MathML source Literature. The problems of price selection and incentive-compatibility in priority queues were dealt with in Mendelson and Whang [10]. They consider the special case in which the penalty functions––which can be interpreted as minus the utility functions––are linear in the mean delays. Conditions (1) become View the MathML source In [10] prices are derived which are optimal and incentive compatible: the prices maximize the system’s ‘net value’, where the choice what class to join is left to the individual users (and the solution is a Nash equilibrium). Importantly, [10] shows that the optima for dedicated classes and open classes coincide. We believe that some aspects of the model of [10] do not apply to the situation of competing data and voice users described above. In the first place, clearly the choice of the penalty functions in [10] is restrictive. As argued above, for low values of the delay the delay-sensitive voice users have a higher utility than the delay-tolerant, whereas for high delay the opposite holds. This cannot be modeled in the framework of [10], as it is not clear whether vd should be larger than vv or vice versa. In other words, the utility curves (and hence the penalty functions) should not have a monotonous relation: they should intersect. We believe that some aspects of the model of [10] do not apply to the situation of competing data and voice users described above. In the first place, clearly the choice of the penalty functions in [10] is restrictive. As argued above, for low values 232 M. Mandjes / Computer Networks 42 (2003) 231–249 of the delay the delay-sensitive voice users have a higher utility than the delay-tolerant, whereas for high delay the opposite holds. This cannot be modeled in the framework of [10], as it is not clear whether vd should be larger than vv or vice versa. In other words, the utility curves (and hence the penalty functions) should not have a monotonous relation: they should intersect. Another interesting approach to service differentiation can be found in Odlyzko [11,12]: he proposes to offer multiple qualities by using multiple logically separated networks with different prices. The idea is that the expensive network attracts the delay-sensitive users, whereas the delaytolerant users opt for the cheap network. Using game-theoretic techniques, Gibbens et al. [2] argues that this mechanism, known as Paris Metro Pricing, does not work if there are multiple competing providers: in order to maximize profits the providers rather focus on one user type. Principles behind congestion pricing are given in, e.g., [3,6]; the former reference explicitly covers heterogeneous users. There are many references with more practical reflections on pricing in multiservice networks, see for instance [1,15], and several articles in [8]. Contribution and organization. This paper looks at the situation in which the utility curves do intersect: for ED 2 ð0; 1Þ it holds that uvðEDÞ > udðEDÞ, whereas for ED > 1 the opposite holds: udðEDÞ > uvðEDÞ. First we look at the situation in which there are large populations of
potential voice and data users sharing a FIFO queue. We see that, depending on the value of the link speed l, the network population will consist of just one class. For small l (i.e., the link is relatively slow) data will dominate, whereas for fast links voice will push aside data. This situation is considered in Section 2. We focus on prices that maximize the providers profit, which is slightly different from the
net value maximization problem solved in [9,10] (cf. social welfare maximization). An important conclusion of our paper is that under our utility curves the solutions of the openclasses model and the dedicated-classes model do not coincide (which did hold in the setting of [10]). Section 3 analyzes the profit maximization problem for the model with dedicated classes, whereas Section 4 focuses on the situation with open classes. As could be expected, Section 4 is more involved: the customers have more options, and therefore the incentive-compatibility requirement is more involved. We find that, depending on the value of the link rate l, different regimes are optimal: for small l only data users will be present, for moderate l the high-priority class is used by voice and the low-priority class by data, whereas for large l voice users dominate. Strikingly, even in the cases where only one type of traffic is present (i.e., small and large l), it is optimal (i.e., profit maximizing) to use both the high-priority and low-priority queue. In other words, even for homogeneous users it is beneficial to introduce service differentiation (and price differentiation). This somewhat counterintuitive result is further explained in Section 5. This section also contains a discussion on the specific shape of the utility function, as well as a numerical example. The paper is concluded by a model in which throughput (rather than packet delay) is the main performance measure. We consider a stream of jobs that is served according to the processor sharing discipline [5, chapter IV]. For a job of given size x, we can (given the load of the queue and the service speed) compute the required transmission time, and hence the throughput during the transmission. The utility is an increasing function of the throughput; we assume that the utility curve UxðÞ is parametrized by the job size x. Section 6 analyzes the situation in which a volume charge is imposed on the jobs (i.e., a fixed price per byte). It shows that under specific assumptions on the ordering of the utility curves, it is beneficial to discriminate the jobs on the basis of their size: if UxðÞ decreases in x, the small jobs (usually referred to as web mice) are preferred over the larger jobs (elephants).