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Demand Curve A (P) Service Requirement g Delay Cost c 1 Atomistic Impatient Customers X 11 '21 12 Random Order Streams Demand Curve 2(P) Service Requirement 1.1. 2 Delay Cost c 2 Capacity g Capacity g P = Price + Expected Delay Cost Figure 1: Two Firms Competing for Two Impatient Customer Segments 5
tion, for example a distribution of customers over the value they derive from an order (see, e.g., Mendelson and Whang (1990)). The assumption that P. is decreasing is natural and expresses the fact that the higher the price, the lower the total demand. For the market to be in equilibrium, the full prices at all firms must be the same, because otherwise customers would switch between firms: Pn(An) =- Pin + CWn(Ail • • • 7 AiN; f); 2 = 1, 2, (1) where f specifies a scheduling policy employed by the firm and is explained below. Wn denotes the expected throughput time of a type n customer at firm i, reflecting the different scheduling treatment customers of different types experience, and the fact that the function Wn is the same for both firms. Without loss of generality, the groups can be ordered such that the waiting cost to service time ratio decreases with the group number: c.A. > = 1, , (N — 1). This ordering anticipates the static priority scheme that we show below to be optimal (and which is commonly referred to as the "c-p rule"). The subscription process works as follows. Each firm i must simultaneously post N quantities Ail , , AiN , one for each customer group. Prices are then determined by the market within each group, depending on the total industry output for the group and the expected delays. That is, firms engage in a game of quantity competition, for which the next section will characterize the form of the unique Nash equilibrium. Price competition is expected to make firm behavior more aggressive (increase output and decrease prices) and is not discussed here. The decision problem for each firm includes N quantities as decision variables and N inverse demand relationships, taking as given automatic price adjustment achieving the market equilibrium. Each firm must now also choose a scheduling policy. A scheduling policy specifies at each point in time when an arrival occurs or a service is completed (these points in time are called decision points), which job present at the facility will be served next, or whether the facility will be idle until the next arrival or service completion. Each firm must announce its stationary scheduling policy f at the beginning of the period, and then customers can assess what average waiting times result from that scheduling policy. FIFO is an example of a simple scheduling policy. It always applies the rule: "serve the customer next who has been waiting the longest", regardless 6
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* Assistant Professor of Operations Management at INSEAD ...

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