Calculating Cost Distributions of a Multiservice Loss System

Congestion pricing has received lots of attention in scientific discussion. Congestion pricing means that the operator increases prices at the time of congestion and the traffic demand are expected to decrease. In a certain sense, shadow prices are an optimal way of congestion pricing: users are charged shadow prices, i.e., the expectations of future losses because of blocked connections. The shadow prices can be calculated exactly from the Howard equation, but this method is difficult. The paper presents simple approximations to the solution of the Howard equation and a way to derive more exact approximations. If users do not react by lowering their demand, they will receive higher bills to pay. Many users do not react to increased prices but would want to know how the congestion pricing mechanism affects the bills. The distribution of the price of a connection follows from knowing the shadow prices and the probability of a congestion state. There is another interesting distribution. The network produces profit to the operator, or equivalently, blocked connections produce a cost to the operator. The average cost rate can be calculated from the Howard equation, but the costs have some distribution. The distribution gives the risk that the actual costs exceed the average costs, and the operator should include this risk in the prices. The main result of this paper shows how to calculate the distribution of the costs in the future for congestion pricing by shadow prices and for congestion pricing with a more simple pricing scheme that produces the same average costs.