Monotonicity and supermodularity results for the Erlang loss system

For the Erlang loss system with s servers and offered load a, we show that: (i) the load carried by the last server is strictly increasing in a; (ii) the carried load of the whole system is strictly supermodular on f(s; a)js = 0; 1; : : : and a > 0g

Capacity expansion for a loss system with exponential demand growth

We study a loss system to forecast the demand for capacity based on the forecast demand for service and a specified service level. A little-used property of the Erlang loss formula allows the linear transformation of demand for service into demand for capacity. Next, given the forecast demand for capacity, we approximate a long-run optimal capacity expansion policy by optimizing over successively longer finite time horizons. Analytical formulas together with regression analysis show the significance of the number of potential customers, frequency and duration of their requests for service, and the specified service level on the demand for capacity. Numerical sensitivity analysis exposes the effects of cost parameters, the demand growth rate and the required rate of return on the optimal time intervals between expansions

Optimal static pricing for a tree network

We study the static pricing problem for a network service provider in a loss system with a tree structure. In the network, multiple classes share a common inbound link and then have dedicated outbound links. The motivation is from a company that sells phone cards and needs to price calls to different destinations. We characterize the optimal static prices in order to maximize the steady-state revenue. We report new structural findings as well as alternative proofs for some known results. We compare the optimal static prices versus prices that are asymptotically optimal, and through a set of illustrative numerical examples we show that in certain cases the loss in revenue can be significant. Finally, we show that static prices obtained using the reduced load approximation of the blocking probabilities can be easily obtained and have near-optimal performance, which makes them more attractive for applications.Massachusetts Institute of Technology. Center for Digital BusinessUnited States. Office of Naval Research (Contract N00014-95-1-0232)United States. Office of Naval Research (Contract N00014-01-1-0146)National Science Foundation (U.S.) (Contract DMI-9732795)National Science Foundation (U.S.) (Contract DMI-0085683)National Science Foundation (U.S.) (Contract DMI-0245352

Evaluating Opportunistic Multi-Channel MAC: Is Diversity Gain Worth the Pain?

We evaluate the performance of an opportunistic multi-channel medium access control protocol and compare it to that of the corresponding single-channel MAC (S-MAC) and a non-opportunistic multi-channel MAC (M-MAC). We do this in three different settings: (1) an ideal scenario where no control channel is used and no sensing delay is incurred, (2) a more realistic scheme where users compete for access on a control channel using random access, and (3) a scheme similar to (2) but with a time-division multiplexing (TDM) based access scheme on the control channel. Our analysis and numerical results show that in terms of delay performance, the random access and competition on the control channel, which typically occupy a fraction of the total bandwidth, almost always wipe out the channel diversity gain, a main motivation behind an opportunistic multichannel MAC. On the other hand opportunistic access increases bandwidth utilization which reduces the system’s total busy time. As a result it helps reduce power consumption in general. When TDM is employed on the control channel, the data sub-channel sensing delay becomes the main bottleneck to attaining better performance. In this case the performance of opportunistic multichannel MAC gets closer to that of the single-channel MAC when the channel sensing overhead is substantially reduced

Analysis of resource pooling games via a new extension of the Erlang loss function

We study a situation where several independent service providers collaborate by pooling their resources into a joint service system. These service providers may represent such diverse organizations as hospitals that pool intensive care beds and ambulances, airline companies that share spare parts, or car rental agencies that pool rental cars. We model the service systems as Erlang loss systems that face a fixed cost rate per server and penalty costs for lost customers. We examine the allocation of costs of the pooled system amongst the participants by formulating a cooperative cost game in which each coalition optimizes the number of servers. We identify a cost allocation that is in the core of this game, giving no subset of players an incentive to split off and form a separate pooling group. Moreover, we axiomatically characterize this allocation rule and show that it can be reached through a population monotonic allocation scheme. To obtain these results, we introduce a new extension of the classic Erlang loss function to non-integral numbers of servers and establish several of its structural properties

On a Bicriterion Server Allocation Problem for a Multidimensional Erlang Loss System

In this work an optimization problem on a classical elementary stochastic system system, modeled as an Erlang-B (M/M/x) loss system, is formulated by using a bicriteria approach. The problem is focused on the allocation of a given total of k servers to a number of groups of servers capable of carrying certain offered traffic processes assumed as Poissonian in nature. Two main objectives are present in this formulation. Firstly a criterion of equity in the grade of service, measured by the call blocking probabilities, entails that the absolute difference between the blocking probabilities experienced by the calls in the different service groups must be as small as possible. Secondly a criterion of system economic performance optimization requires the total traffic carried by the system, to be maximized. Relevant mathematical results characterizing the two objective functions and the set N of the non-dominated solutions, are presented. An algorithm for traveling on N based on the resolution of single criterion convex problems, using a Newton-Raphson method, is also proposed. In each iteration the two first derivatives of the Erlang-B function in the number of circuits (a difficult numerical problem) are calculated using a method earlier proposed. Some computational results are also presented

Coordination of staffing and pricing decisions in a service firm

Customer demand is sensitive to the price paid for the service in many service environments. Using queueing theory framework, we develop profit maximization models for jointly determining the price and the staffing level in a service company. The models include constraints on the average waiting time and the blocking probability. We show convexity of the single-variable subproblem under certain plausible assumptions on the demand and staffing cost functions. Using numerical examples, we investigate the sensitivity of the price and the staffing level to changes in the marginal service cost and the user-specified constraint on the congestion measure. Copyright © 2008 John Wiley & Sons, Ltd

The Power of Static Pricing for Reusable Resources

We consider the problem of pricing a reusable resource service system. Potential customers arrive according to a Poisson process and purchase the service if their valuation exceeds the current price. If no units are available, customers immediately leave without service. Serving a customer corresponds to using one unit of the reusable resource, where the service time has an exponential distribution. The objective is to maximize the steady-state revenue rate. This system is equivalent to the classical Erlang loss model with price-sensitive customers, which has applications in vehicle sharing, cloud computing, and spare parts management. Although an optimal pricing policy is dynamic, we provide two main results that show a simple static policy is universally near-optimal for any service rate, arrival rate, and number of units in the system. When there is one class of customers who have a monotone hazard rate (MHR) valuation distribution, we prove that a static pricing policy guarantees 90.4\% of the revenue from the optimal dynamic policy. When there are multiple classes of customers that each have their own regular valuation distribution and service rate, we prove that static pricing guarantees 78.9\% of the revenue of the optimal dynamic policy. In this case, the optimal pricing policy is exponentially large in the number of classes while the static policy requires only one price per class. Moreover, we prove that the optimal static policy can be easily computed, resulting in the first polynomial time approximation algorithm for this problem

Energy Portfolio Optimization of Data Centers

Data centers have diverse options to procure electricity. However, the current literature on exploiting these options is very fractured. Specifically, it is still not clear how utilizing one energy option may affect selecting other energy options. To address this open problem, we propose a unified energy portfolio optimization framework that takes into consideration a broad range of energy choices for data centers. Despite the complexity and nonlinearity of the original models, the proposed analysis boils down to solving tractable linear mixed-integer stochastic programs. Using experimental electricity market and Internet workload data, various insightful numerical observations are reported. It is shown that the key to link different energy options with different short- and long-term profit characteristics is to conduct risk management at different time horizons. Also, there is a direct relationship between data centers' service-level agreement parameters and their ability to exploit certain energy options. The use of on-site storage and the deployment of geographical workload distribution can particularly help data centers in utilizing high-risk energy choices, such as offering ancillary services or participating in wholesale electricity markets