UNOBSERVABLE QUEUES
5. Customers know their demand
Consider the model of Section 1 but with one change: customers base their decision of whether to join the queue on their service time. We assume that the service time is private information of the customer. We consider equilibrium threshold strategies under three service disciplines: FCFS, egalitarian processor sharing (EPS), and shortest service first. In all cases, the equilibrium behavior is based on a threshold, xe, such
that only customers with service requirementt≤xe join. By the same
reasoning as in Section 1, the socially optimal threshold, x∗ is smaller thanxe.
Assume that the service time is a continuous random variable with distribution and density functions G and g, respectively, and that Λ is the potential arrival rate. Given that only customers with service duration of at mostx join, the arrival rate isλ(x) = ΛG(x).
5.1.
FCFS
Suppose that only customers with a service requirement of at most
x join. Then, the density function of service requirements among cus- tomers who join is Gg((yx))for 0≤y≤x(and 0 elsewhere), and the expected service requirement among them is
m(x) = 1 G(x) Z x y=0 yg(y)dy. Let ρ(x) = ΛG(x)m(x) = Λ Z x y=0 yg(y)dy (3.7)
be the effective utilization factor of the system. By the Khintchine- Pollaczek formula (1.7), the expected queueing time for those who join is9
Wq(x) =
ΛRx
y=0y2g(y)dy
2(1−ρ(x)) .
9Assuming anM/M/1 model rather thanM/G/1 does not lead to a simpler analysis: the
Unobservable queues 59 The equilibrium thresholdxe is defined by
C(xe+Wq(xe)) =R,
whereas the social optimal thresholdx∗ is
x∗ = arg max
x {ΛG(x)(R−C[m(x) +Wq(x)])}.
5.2.
EPS
The EPS, model was solved by Haviv [75]. In an M/G/1 EPS queue with a utilization factorρ, the expected time in the system for a customer with service requirementt is (see, for example, [148] p. 174)
Wt=
t
1−ρ.
When the threshold strategyx is applied, the expected time in the sys- tem for a customer with service requirement t (t > x is possible here) is
Wt(x) =
t
1−ρ(x),
whereρ(x) is as defined in (3.7). The equilibrium threshold xe is given
by
Cxe
1−ρ(xe)
=R.
Using (1.3), the expected number of customers in the system, under the thresholdx, is ρ(x) 1−ρ(x). Hence, x∗= arg max x ΛG(x)R− Cρ(x) 1−ρ(x) ,
and by the first-order conditions,x∗ satisfies
Cx∗
[1−ρ(x∗)]2 =R.
It is possible to regulate the system by imposing an admission fee of
T such that
R−T = Cx∗ 1−ρ(x∗),
or a fee of p per unit of service time such that px∗ =T, or a fee t per unit of time in the system such that
R= (C+t)x∗ 1−ρ(x∗).
60 TO QUEUE OR NOT TO QUEUE
5.3.
Shortest service first
Suppose now that the service requirement of an arrival is also known to the server, and that the service regime is such that customers with shorter service requirements receive preemptive priority over customers with longer requirements. The same rule applies to the order in which preempted customers return to service. Note that priority levels are based on original requirements and not on the residual requirements which change while in the system. Service is resumed from the point where it was interrupted.
Customers know their demand and need to decide irrevocably whether or not to join. Consider an equilibrium strategy of the threshold type: for some xe, all those with demand x ≤ xe join, whereas the others
balk. Denote by x∗ the socially optimal threshold. We observe that the customer with the longest service requirement who joins imposes no externalities on the others. Therefore, his objective coincides with the social objective, and we conclude that he joins if and only if this is socially desired. Thus, we conclude thatxe=x∗.
We next show how to compute this common threshold. LetTxdenote
the expected waiting time of a customer whose service time is x, when the threshold strategy x is used by all. Then (see, for example [89] p. 124), Tx= x(1−ρ(x)) + ΛRx y=0g(y)y dy [1−ρ(x)]2 Txe uniquely satisfies CTxe =R.
6.
Finite buffer
Lin and Ross [108] considered a queueing system with a waiting area (or buffer) of bounded size. In their model, arrivals are forced to balk when the buffer is full. Since admission to the queue is not guaranteed, we refer to an arrival as atrial.10
To illustrate such models, Lin and Ross considered anM/M/1/1 sys- tem, namely a single server system with no waiting room. The service value isR, and without loss of generality assume that the time value is
C= 0.11 There is also a cost T < Rassociated with each trial. This is a real cost, not a transfer payment. The potential arrival rate is denoted by Λ, and letρ= Λµ. The customer’s problem in this model is: to try or not to try.
10A model where rejected customers retry later is discussed in§6.4. 11Otherwise, just replaceRwithR−C
Unobservable queues 61 A strategy is characterized by the probability pthat a customer tries. The effective trial rate is thenλ=pΛ and the server is idle with proba- bility µ+µpΛ. The (expected) payoff for one who tries is therefore
Rµ
µ+pΛ−T. (3.8)
Clearly, this is an ATC situation and therefore there exists a unique equilibrium. By (3.8), if Λ< µR−TT, then regardless of what the others do, one’s best action is to try (in other words, trying is a dominant strategy). Otherwise, no dominant strategy exists. If p = µRT−ΛT then a customer is indifferent between trying or not. Hence, trying with this probability is the equilibrium strategy. Denote the equilibrium trial probability bype, then pe= 1 ρ≤ R−TT R−T T ρ ρ > R−TT .
The socially optimal trial probability is defined by
p∗ = arg max p p Rµ µ+pΛ −T . Hence, p∗= 1 ρ≤qRT −1 pR T−1 ρ ρ > q R T −1 .
When ρ≤qRT −1,pe=p∗ = 1. Otherwise,pe> p∗. It is possible to
inducep∗ in equilibrium by imposing an appropriate fee on trials or on service completions.
Sumita, Masuda and Yamakawa [166] considered a system with a finite buffer of sizeK. A customer who encounters a full buffer is rejected and obtains service from an alternative server. If the buffer is not full then the customer is accepted. To describe their model we define the following functions:
V(λ) - aggregate utility gained by the arriving customers if all are accepted.
R(λ) - aggregate utility gained by the arriving customers if all are rejected.
62 TO QUEUE OR NOT TO QUEUE β(λ, µ, K) - probability of rejection.
G(λ, µ, K) - expected cost incurred by accepted customer.
M(λ, µ, K) - expected cost incurred by rejected customer.
pA - fee imposed on accepted customers.
pR - fee imposed on rejected customers.
The social objective is to maximizeαV +βR−λ(αG+βM) subject to the equilibrium condition
αV′+βR′ =α(pA+G) +β(pR+M).
An arriving customer causes negative externalities in two ways: by in- creasing waiting costs of other customers, and by increasing the prob- ability of rejection for future customers. The authors call the latter type loss externalities. They proved that the equilibrium arrival rate, when no admission fees are imposed, is higher than the optimal arrival rate, and showed that this can be corrected by appropriate fees. They also solved the long-run problem in which both µ and K are decision variables and a cost c(µ, K) is added to the social objective function.