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We develop a framework for asymptotic optimization of a queueing system. The motivation is the sta ng problem of call centers with 100's of agents or more. Such a call center is modeled as an M M N queue, where the number of agents N is large. Within our framework, we determine the asymptotically optimal sta ng level N that trades o agents' costs with service quality: the higher the latter, the more expensive is the former. As an alternative t o this optimization, we also develop a constraint satisfaction approach where one chooses the least N that adheres to a given constraint o n w aiting cost. Either way, the analysis gives rise to three regimes of operation: quality-driven, where the focus is on service quality; e ciencydriven, which emphasizes agents' costs; and a rationalized regime that balances, and in fact uni es, the other two. Numerical experiments reveal remarkable accuracy of our asymptotic approximations: over a wide range of parameters, from the very small to the extremely large, N is exactly optimal, or it is accurate to within a single agent. We demonstrate the utility o f our approach b y revisiting the square-root safety sta ng principle, which is a long-existing ruleof-thumb for sta ng the M M N queue. In its simplest form, our rule is as follows: if c is the hourly cost of an agent, and a is the hourly cost of customers' delay, then N = R + y a c p R, where R is the o ered load, and y is a function that is easily computable.2000 Mathematics Subject Classi cation: 60K25 primary, 90B22 secondary.

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsUser-level performance of channel-aware scheduling algorithms in wireless data networks Channel-aware scheduling strategies, such as the Proportional Fair algorithm for the CDMA 1xEV-DO system, provide an effective mechanism for improving throughput performance in wireless data networks by exploiting channel fluctuations. The performance of channel-aware scheduling algorithms has mostly been explored at the packet level for a static user population, often assuming infinite backlogs. In the present paper, we focus on the performance at the flow level in a dynamic setting with random finite-size service demands. We show that in certain cases the user-level performance may be evaluated by means of a multi-class Processor-Sharing model where the total service rate varies with the total number of users. The latter model provides explicit formulas for the distribution of the number of active users of the various classes, the mean response times, the blocking probabilities, and the mean throughput. In addition we show that, in the presence of channel variations, greedy, myopic strategies which maximize throughput in a static scenario, may result in sub-optimal throughput performance for a dynamic user configuration and cause potential instability effects.

The relative delay tolerance of data applications, together with the bursty traffic characteristics, opens up the possibility for scheduling transmissions so as to optimize throughput. A particularly attractive approach, in fading environments, is to exploit the variations in the channel conditions, and transmit to the user with the currently 'best' channel. We show that the 'best' user may be identified as the maximum-rate user when the feasible rates are weighed with some appropriately determined coefficients. Interpreting the coefficients as shadow prices, or reward values, the optimal strategy may thus be viewed as a revenue-based policy, which always assigns the transmission slot to the user yielding the maximum revenue. Calculating the optimal revenue vector directly is a formidable task, requiring detailed information on the channel statistics. Instead, we present adaptive algorithms for determining the optimal revenue vector on-line in an iterative fashion, without the need for explicit knowledge of the channel behavior. Starting from an arbitrary initial vector, the algorithms iteratively adjust the reward values to compensate for observed deviations from the target throughput ratios. The algorithms are validated through extensive numerical experiments. Besides verifying long-run convergence, we also examine the transient performance, in particular the rate of convergence to the optimal revenue vector. The results show that the target throughput ratios are tightly maintained, and that the algorithms are well able to track sudden changes in the channel conditions or throughput targets.2000 Mathematics Subject Classification: 60K25 (primary), 68M20, 90B18, 90B22 (secondary).

We consider a system of N parallel single-server queues with unit exponential service rates and a single dispatcher where tasks arrive as a Poisson process of rate λ(N). When a task arrives, the dispatcher assigns it to a server with the shortest queue among d(N) randomly selected servers (1 d(N) N). This load balancing strategy is referred to as a JSQ(d(N)) scheme, marking that it subsumes the celebrated Join-the-Shortest Queue (JSQ) policy as a crucial special case for d(N) = N.We construct a stochastic coupling to bound the difference in the queue length processes between the JSQ policy and a JSQ(d(N)) scheme with an arbitrary value of d(N). We use the coupling to derive the fluid limit in the regime where λ(N)/N → λ < 1 as N → ∞ with d(N) → ∞, along with the associated fixed point. The fluid limit turns out not to depend on the exact growth rate of d(N), and in particular coincides with that for the JSQ policy. We further leverage the coupling to establish that the diffusion limit in the critical regime where (N − λ(N))/ √ N → β > 0 as N → ∞ with d(N)/( √ N log(N)) → ∞ corresponds to that for the JSQ policy. These results indicate that the optimality of the JSQ policy can be preserved at the fluid-level and diffusion-level while reducing the overhead by nearly a factor O(N) and O( √ N/ log(N)), respectively. * d.mukherjee@tue.nl

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsUser-level performance of channel-aware scheduling algorithms in wireless data networks Channel-aware scheduling strategies, such as the Proportional Fair algorithm for the CDMA 1xEV-DO system, provide an effective mechanism for improving throughput performance in wireless data networks by exploiting channel fluctuations. The performance of channel-aware scheduling algorithms has mostly been explored at the packet level for a static user population, often assuming infinite backlogs. In the present paper, we focus on the performance at the flow level in a dynamic setting with random finite-size service demands. We show that in certain cases the user-level performance may be evaluated by means of a multi-class Processor-Sharing model where the total service rate varies with the total number of users. The latter model provides explicit formulas for the distribution of the number of active users of the various classes, the mean response times, the blocking probabilities, and the mean throughput. In addition we show that, in the presence of channel variations, greedy, myopic strategies which maximize throughput in a static scenario, may result in sub-optimal throughput performance for a dynamic user configuration and cause potential instability effects.

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsWireless Data Performance in Multi-Cell Scenarios ABSTRACT The performance of wireless data systems has been extensively studied in the context of a single base station. In the present paper we investigate the flow-level performance in networks with multiple base stations. We specifically examine the complex, dynamic interaction introduced by the strong impact of interference from neighboring base stations. We derive two types of lower and upper bounds for the number of active flows, transfer delays and flow throughputs in the various cells. While the first type of bounds are rather rough and simple to compute, the second type of bounds are sharper, but harder to calculate. In order to obtain closed-form estimates for the latter bounds, we introduce two limit regimes, termed fluid and quasi-stationary regime, where the system dynamics evolve on a very fast and a very slow time scale, respectively. Importantly, the performance in both limit regimes is insensitive, thus yielding simple, explicit estimates that render the detailed statistical characteristics of the system largely irrelevant. Numerical experiments show that the upper bounds evaluated in the quasi-stationary regime provide conservative and extremely tight approximations. Mathematics Subject Classification: 60K25, 68M20Keywords and Phrases: wireless data networks, elastic traffic, insensitivity, state-dependent multi-class processor sharing, time-varying service, bounds, fluid regime, quasi-stationary regime, stability, throughput. Note: The work of the second and third authors was carried out in part under the project PNA2.2 ``Wireless Networks'' and supported by a grant from France Telecom R&D. Wireless Data Performance in Multi-Cell Scenarios ABSTRACTThe performance of wireless data systems has been extensively studied in the context of a single base station. In the present paper we investigate the flow-level performance in networks with multiple base stations. We specifically examine the complex, dynamic interaction introduced by the strong impact of interference from neighboring base stations. We derive two types of lower and upper bounds for the number of active flows, transfer delays and flow throughputs in the various cells. While the first type of bounds are rather rough and simple to compute, the second type of bounds are sharper, but harder to calculate. In order to obtain closed-form estimates for the latter bounds, we introduce two limit regimes, termed fluid and quasi-stationary regime, where the system dynamics evolve on a very fast and a very slow time scale, respectively. Importantly, the performance in both limit regimes is insensitive, thus yielding simple, explicit estimates that render the detailed statistical characteristics of the system largely irrelevant. Numerical experiments show that the upper bounds evaluated in the quasi-stationary regime provide conservative and extremely tight app...

International audienceOver the past few years, the design and performance of channel-aware scheduling strategies have attracted huge interest. In the present paper, we examine a somewhat different notion of scheduling, namely coordination of transmissions among base stations, which has received little attention so far. The inter-cell coordination comprises two key elements: (i) interference avoidance and (ii) load balancing. The interference avoidance involves coordinating the activity phases of interfering base stations so as to increase transmission rates. The load balancing aims at diverting traffic from heavily loaded cells to lightly loaded cells. Numerical experiments demonstrate that inter-cell scheduling may provide significant capacity gains