Price and priority competition

Lederer and Li [101] considered a competitive multi-server extension of the model of Mendelson and Whang [124] (_§4.4.3). In their model, cus- tomers belong toclasses. Service of customers of classz(“z-customers”), is done at a rate ofµz, and their time value isCz. Servers can discrimi-

nate among customers by their class. Serveri selects a priority rulefi,

and chargesz-customers a pricepi(z).Denote byλi= (λi(z)) the vector

of arrival rates to queueiwhose priority rule isfi. Az-customer selects a

server to minimize his expected full pricePi(z) =pi(z) +CzWi(z, λi, fi),

where Wi(z, λi, fi) is the expected waiting time of a z-customer who

joins queuei. Denote the server’s cost when serving the arrival rates of

λ_i byci(λi).

Lederer and Li assumed that the number of servers is large so that each server’s influence on the equilibrium full prices is negligible. They also assumed that entry of new servers is not possible (a short-term model).

In equilibrium, the servers withλi(z)>0 have the same expected full

pricePz forz-customers:

Pz =pi(z) +CzWi(z, λi, fi).

Therefore, the profit that serverigets from servingz-customers can be written as

λi(z)pi(z) =λi(z)[Pz−CzWi(z, λi, fi)].

7_{The two conditions also imply lower and upper bounds onp} 2.

Competition among servers 149 This equation also holds whenλi(z) = 0, since in this case both its sides

are 0. Hence, the total profit of serveriis Πi(P , λi, fi) =

X

z

λi(z)[Pz−CzWi(z, λi, fi)]−ci(λi), (7.5)

whereP = (Pz).

Following [124], Lederer and Li considered the server’s problem as that of maximizing Πi(P , λi, fi) by selecting a vector of arrival ratesλi

and queue disciplinesfi.

Applying Little’s formula to (7.5) we obtain that for server i, i = 1, . . . , n,

Πi(P , λi, fi) =

X

z

[Pzλi(z)−CzLi(z, λi, fi)]−ci(λi), (7.6)

where Li(z, λi, fi) is the expected number of z-customers at the i-th

queue.

For any given λ_i, (7.6) is maximized when the service discipline min- imizes P

zCzLi(z, λi, fi). It is well known that under fairly general

conditions, the minimizing discipline is that which gives higher prior- ity to customer classes with higherCzµzvalues (theCµ-rule) (see [52]).

Therefore, this rule will be adopted by all servers. The variable fi can

now be omitted, and thei-th server’s problem is to chooseλ_ito maximize Πi(P , λi) as in (7.5), for any givenP.

Suppose that class z has a demand function dso that given the full pricePz, the arrival process ofz-customers is Poisson with rated(z, P).

An equilibrium is a set of vectors P and λ_i for every serveri, such that

λ_i solves server i’s profit-maximizing problem, given P and d(z, P) = P

iλi(z) for every class z.

Lederer and Li proved the following results: IfP

zCzLi(z, λi) +ci(λi) is strictly convex inλi for everyi, then an

equilibrium exists.

Equilibria are incentive-compatible.

If there is just one class of customers then there exists a unique equilibrium.

Suppose that the service requirements of all classes have an identical exponential distribution, and that for every i, ci(λi) = ci[Pzλi(z)].

150 TO QUEUE OR NOT TO QUEUE

5.

Search among competing servers

Davidson’s [40] model is a combination of the generic models of ob- servable and unobservable queues. Davidson considered a large number of servers so that the queue lengths at these servers are assumed to be independent. Customers have heterogeneous time values and every customer knows his own value. Balking is not allowed, and customers search among the servers until they decide to join.

The arrival rate is λ per server. The cost incurred by a customer with time valueC (a “C-customer”) due to an inspection of a queue is

b+wC, where band w are constant across customers (w is interpreted as the search time). The goal of a customer is to minimize his expected costs due to search, admission, and waiting.

A C-customer’s strategy is defined by a threshold B(C) on his full price. He joins a server who charges p and whose queue size isnif and only if his expected full price is at most B(C), that is, if and only if

p+Cn+1_{µ ≤}B(C).

Incomplete information Assume that customers are uninformed on the prices charged by the servers. Thus, by visiting a server the customer learns both its price and its queue length.

Since customers do not know the prices in advance, the arrival rate to a server is independent of its price. After learning the price and observing the queue, an arriving customer decides whether to join or to continue searching. Being aware of the customers’ decision process, each server selects a price that optimizes its expected profits. The arrival rate to a server is independent of its price, and we are looking for a symmetric equilibrium under which all servers select the same price.

Davidson didn’t specify how to compute the equilibrium threshold functionB(C). We now briefly outline this process. Note that since balking is not allowed in this model, the customers only optimize their search and waiting costs, and the equilibrium search strategies are independent of the prices. If a price p is required by all servers, this is a fixed cost that every customer must pay, and the equilib- rium threshold full price will be increased by p. Let B0(C) be the

equilibrium threshold function, given that p = 0. Let qi denote the

steady state probability of i customers at a random server, when

B0(C) is adopted by all customers. Let AC(β) denote the expected

cost incurred by aC-customer who chooses a thresholdβ, given that everybody else follows B0(C). Let i(β, C) = max{i:C(i+ 1)≤βµ}

Competition among servers 151

β still joins, given that the admission fee is 0. Then,

AC(β) =b+wC+ X i≤i(β,C) qiC i+ 1 µ +AC(β)P r[i > i(β, C)], or AC(β) = b+wC+P i≤i(β,C)qiCi+1_µ P r[i_≤i(β, C)] .

One should find from this relation the best response, and then the equilibrium function B0(C). In the next stage, the equilibrium price

can be found by

pe= arg max_p {p

Z

C

P r[i_≤i(B0(C) +pe−p, C)]dC}.

Complete information In this case the prices are advertised by the servers, and customers search among them to reveal their queue lengths. Davidson considered a simplified model in which a propor- tion α of the customers has C = 0, and the rest have C = 1. Note that the assumption that one of the cost values is 0, is part of the model, whereas the assumption that the other cost value is 1, is with- out loss of generality. The respective thresholds are denoted by B0

and B1. 0-customers are indifferent to the queue length, and hence

join only those servers that offer the lowest price, p0. 1-customers

may join servers with different prices, as long as the expected full price at these servers is the same. Davidson concluded that: “Since consumers do not care which type of server they end up at, it seems logical to assume that they will choose the server randomly, and all servers will face the same arrival rate.” This conclusion is problem- atic since the fact that customers are indifferent in regard to a set of servers, does not mean that their arrival rates are arbitrary (see Re- mark 1.1). The rates should be set at those levels which make their expected full prices equal. It is an open problem to continue David- son’s research. Another seemingly harder open problem is a model that assumes a small number of servers so that the queue lengths are not independent.