In this paper, we investigate optimal load balancing strategies for a multi-class multi-server processor-sharing system with a Poisson input stream, heterogeneous service rates, and a server-dependent holding cost per unit time. Specifically, we study (i) the centralized setting in which a dispatcher routes incoming jobs based on their service time requirements so as to minimize the weighted mean sojourn time in the system; and (ii) the decentralized, distributed non-cooperative setting in which each job, aware of its service time, selects a server with the objective of minimizing its weighted mean sojourn time in the system. For the decentralized setting we show the existence of a potential function, which allows us to transform the non-cooperative game into a standard convex optimization problem. For the two aforementioned settings, we characterize the set of optimal routing policies and obtain a closed form expression for the load on each server under any such policy. Furthermore, we show the existence of an optimal policy that routes a job independently of its service time requirement. We also show that the set of servers used in the decentralized setting is a subset of set of servers used in the centralized setting. Finally, we compare the performance perceived by jobs in the two settings by studying the so-called Price of Anarchy (PoA), that is, the ratio between the decentralized and the optimal centralized solutions. When the holding cost per unit time is the same for all servers, it is known that the PoA is upper bounded by the number of servers in the system. Interestingly, we show that the PoA for our system can be unbounded. In particular this indicates that in our system, the performance of selfish routing can be extremely inefficient.

No abstract

Abstruct-We study an Adaptive Window Protocol (AWP) witha general increase and decrease profile in the presence of window dependent random losses. We derive a steady-slate Kolmogorov equation and obtain its solution in analytic form. We obtain some stochastic ordering relations for a protocol with different bounds on window, A closed form necesary ahd sufticient stability condition using the stochastic ordering for the window process is established. Finally, we apply the general results to particular TCP versions such as New Reno TCP, Scalable TCP and Highspeed TCP. We observe that HighSpeed TCP can be used to approximate almost any kind of window behavior by varying only one design parameter. I . INTRODUCTIOYMost of the performance studies of Adaptive Window Protocols (AWP) consider specific instances of the problem (for example [3]. 121, [61 study Additive Increase Multiplicative Decrease (ATMD) protocols). However, various modifications to TCP are frequently proposed to address specific problems arising in various types of networks; recent examples include Highspeed TCF [51 and Scalable TCP C41 proposed for very high bandwidth-delay product networks. These new proposals can also be viewed in the framework of Additive Increase protocols so that now the additive increase in a round-trip time is function of the current window size (it is constant in the case of standard TCP). Performance related analysis of any such protocol has always been an important issue. It is thus desirable to have a general framework (and its solution) for performance anakysis of an AWP.The loss process seen by a TCP sender may have its origin in deliberate markingldropping owing to some active queue management (AQM) scheme employed in the network, or could be due to congestion losses or link errors, in general the rate of receiving a loss signal will depend on the window process itself (see [6] for related discussion). In this study, we consider a general state dependent loss rate. It is clear that the stability of window process of a general AWP will depend on the rate at which it receives loss signals. For example, an aggressive protocol may result in very high windows for moderate loss rates and vice versa.Stability of the window process is thus interesting to study.We address the problem of finding conditions for stability of a general AWP controlled window evolution under a general statedependent loss rate. The contributions (and organization) of this work is as follows:Section 11: We give a characterization of a general AWP and identify the various quantities that determine the performance o f such protocols. The window evolution under a general AWP is mapped to that under an AWP with a linear increase profile (like in standard TCP). Kolmogorov equations satisfied by the stationary probability measure is then derived. Section 111: Gives conditions under which two AWPs have related stationary distribution. Furthermore, we demonstrate that the window process under multiplicative decrease protocol is also related to the workload proce...