Pricing and Congestion Management in a Network With Heterogeneous Users

We consider an economic model for a communication network with utility-maximizing elastic users who adapt to congestion by adjusting their fl ows. Users are heterogeneous with respect to both the utility they attach to different levels of flow and their sensitivity to delay or other measures of congestion. Following Kelly at al [4], we introduce dynamical rate-control algorithms, based on the users' utility functions and delay sensitivities, as well as tolls charged by the system, and examine their behavior. We show that allowing heterogeneity with respect to delay sensitivity introduces a fundamental non-convexity into the congestion-cost functions. As a result, there are typically multiple stationary points. Hence marginal-cost pricing - equating users' marginal utilities to their marginal costs - may identify a local maximum or even a saddle point, rather than a global maximum. We present a simple example in which the only interior stationary point is a saddlepoint, which is dominated by all the single-user optimal allocations. Thus, heterogeneity of users can lead to class dominance: a situation in which the system is dominated by a single user or class of users under an optimal flow allocation. The dynamical-system rate-control algorithm may converge to a local rather than global maximum, depending on the starting point

Semi-stationary clearing processes

A clearing process describes the net quantity in a service system (e.g. a batch service queue or dam) which receives an exogenous random input over time. and has an output mechanism that intermittently clears random quantities from the system. A semistationary clearing process is strictly stationary over its random clearing epochs. We describe the asymptotic distribution of such processes and show how it arises in limits of certain functionals of these processes. An asymptotic distribution is different from a limiting distribution, but it has some similar properties. We then identify some clearing processes whose asymptotic distribution is uniform. This is true for modulo c clearing with a stationary input if the Palm probability is used rather than the usual probability. Our results on this give a partial answer to an anomaly in the classical economic lot size inventory model. We also present a functional central limit law and law of the iterated logorithm for clearing processes, as well as a result on the convergence of a sequence of such processes.

Individual versus Social Optimization in the Allocation of Customers to Alternative Servers

Customers arrive at a service area according to a Poisson process. An arriving customer must choose one of K servers without observing present congestion levels. The only available information about the kth server is the service time distribution (with expected duration \mu k -1 ) and the cost per unit time of waiting at the kth server (h k). Although service distributions may differ from server to server and need not be exponential, it is assumed that they share the same coefficient of variation. Individuals acting in self-interest induce an arrival rate pattern (\lambda \^ 1, \lambda \^ 2, ..., \lambda \^ k). In contrast, the social optimum is the arrival rate pattern (\lambda 1 *, \lambda 2 *, ..., \lambda k *) which minimizes long-run average cost per unit time for the entire system. The main result is that \lambda \^ k's and \lambda \^ k*'s differ systematically. Individuals overload the servers with the smallest h k/\mu k values. For an exponential service case with pre-emptive LIFO service an alternative charging scheme is presented which confirms that differences between individual and social optima occur precisely because individuals fail to consider the inconvenience that they cause to others.queueing, individual vs social optimization, joining behavior