PRIORITIES
2. Unobservable queues
We now consider the unobservable version of the model discussed in Section 1.2, where customers decide whether to purchase priority at the price of θ without first observing the queue. A strategy is associated with a probabilityp of purchasing priority.
There are two ways in which an increase in the parameter p, used by all other customers, affects a given customer. On the one hand, it increases the expected number of customers who overtake an ordinary customer and hence increases the incentive to purchase priority. On the other hand, it decreases the expected number of ordinary customers that a priority customer expects to overtake and hence decreases the incentive to purchase priority. The latter effect does not exist in the observable model since there, the number of ordinary customers to be overtaken is known at the time of making the decision. Therefore, unlike the observable model, it is not intuitively clear here if the model is of the FTC type. It turns out however, that it is, as proved in the next theorem.
Under the FTC assumption, there are three possibilities for an equi- librium, exactly as in the shuttle model of§1.5: there may be a unique equilibrium with eitherp= 0 orp= 1, or there may be three equilibria,
p= 0,p= 1 and a thirdpesuch that 0< pe<1. The precise conditions
are listed in the following theorem: Theorem 4.5
The model is of the FTC type.
If θ≤ µ(1Cρ−ρ) then p= 1 is a dominant strategy; If θ≥ µ(1Cρ−ρ)2 then p= 0 is a dominant strategy;
If µ(1Cρ−ρ) < θ < µ(1Cρ−ρ)2 then there are three equilibria: p= 0, p= 1 and p= 1ρ−θµ(1C−ρ).
The pure equilibria are ESS. Mixed equilibria are not.
Proof: Given that strategy p is adopted, the arrival process of priority customers is Poisson with rate λp and therefore the expected waiting
84 TO QUEUE OR NOT TO QUEUE time of a priority customer is µ−1λp. LetW denote the expected waiting time of an ordinary customer. Then, the expected waiting time of a random customer can be computed in two ways. First, it equals the expected waiting time in a FCFS queue, which is µ−1λ. Second, it is
1
µ−λp with probability p andW with probability 1−p. Thus,
1
µ−λ = p
µ−λp + (1−p)W,
from which we conclude thatW = (µ−λ)(µµ−λp).5
The reduction, f(p), in expected waiting costs due to becoming a priority customer is
f(p) = λC
(µ−λ)(µ−λp).
This function is monotone increasing withp, from which the FTC prop- erty follows.
Hence,
When θ ≤ f(0), it is uniquely optimal for a customer to purchase priority no matter what the others do. In other words, this is a dominant strategy.
Whenθ≥f(1), it is uniquely optimal for a customer not to purchase priority no matter what the others do. Again, this is a dominant strategy.
Whenf(0)< θ < f(1) there exists a unique valuepe, 0< pe<1, such
that given that pe is adopted by the other customers, a customer is
indifferent between purchasing priority or not. Solvingf(p) =θleads to pe= 1ρ− θµ(1C−ρ). This equilibrium is not ESS: if p is larger than
pe, the unique best response is to buy priority, whereas ifpis smaller
than pe, then the unique best response is not to buy. Therefore,
in addition to the mixed equilibrium, there are two pure equilibria, where all buy priority and where none do. These pure equilibria are ESS.
Figure 4.3 depicts the best response function for the three cases. Remark 4.6 The second item of Theorem 4.5 means that ifθis smaller than the expected queueing cost under the FCFS discipline, then all
Priorities 85 1 1 1 1 1 p p p 45o pe 1 θ > f(1) f(0)< θ < f(1) θ < f(0)
Figure 4.3. Best response vs. fraction of priority customers
customers will purchase priority in equilibrium. Thus, the additional option of buying priority makes everybody worse-off: all customers pay
θ but in practice nobody gains from it. This is an example of rent dissipation, see [94, 169]. Similarly, when f(0)< θ < f(1), a fraction of the customers purchases priority in equilibrium; those who do not purchase priority are worse-off because they are pushed to the back of the queue; those who purchase priority have in equilibrium the same expected net benefit as those who do not, and therefore these customers are also worse-off.
We have seen that in both the observed and the unobserved models the situation is of the FTC type. Agastya [4] suggested a static model which yields a different outcome. Suppose that n+ 1 customers are present in a queue. Tag one of the customers and assume that of the other n
customers x are ordinary customers and n−x are priority customers. The service order in each class is random. If the tagged customer buys priority then his expected queueing time is 2xµ. Otherwise, if he joins as an ordinary customer, his expected queueing time is (x+ n−2x)1µ.
The expected amount saved when buying priority is equal to 2nµ. An interesting outcome is that this saving is independent ofx!
Suppose now that each of the n+ 1 customers possesses the option of payingθ for priority. Then, if θ < C2nµ (respectively, θ > C2nµ) there is a unique equilibrium in which all (respectively, none) buy priority. If
θ=C2nµ then any strategy is an equilibrium.