The Review of Economic Studies Ltd.
On Partial Equilibrium in a Queuing System with Two Servers
Author(s): Israel Luski
Source: The Review of Economic Studies, Vol. 43, No. 3 (Oct., 1976), pp. 519-525
Published by: The Review of Economic Studies Ltd.
Stable URL: http://www.jstor.org/stable/2297230
Accessed: 24/01/2010 07:15
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless
you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you
may use content in the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
http://www.jstor.org/action/showPublisher?publisherCode=resl.
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
The Review of Economic Studies Ltd. is collaborating with JSTOR to digitize, preserve and extend access to
The Review of Economic Studies.
http://www.jstor.org
On
Queuing
Equilibrium in
Partial
System
with
Two
a
Servers
ISRAEL LUSKI
Ben Gurion University
1. INTRODUCTION
Thispaperwill dealwiththe followingquestion: whathappensin a servicesystemwherethe
serviceis suppliedby two competitivefirms? By a servicesystemwe mean any system
wherecustomershave to wait in lines beforethey are served. As we will show, we cannot
apply the usual resultsof economictheoryto servicesystems. For example,in ordinary
economictheorytwo firmssupplyingthe same good will alwayssell it at the same price.
This resultis not alwaystruefor the case of serviceand waitinglines. We will discussthe
conditionsunderwhich two servicefirmssellingat the same price are not in equilibrium.
A discussionof queuingsystemswith singleserver(that is, a monopoly)are found in
the worksof Edelson[1], Naor [3], Knudsen[2] and others. However,thereis no mention
in the literatureof a generalequilibriumin a queuingsystemwith more than one server.
Betweenthese two extremecases, the generalequilibriumand the determinatesingle-firm
model, lie the partialequilibriumcases, and this paperdealswith one of these.
2. TWO FIRMS, TWO PRICES
forces come into play. These
When a good is sold at differentprices,price-equalization
forcesdo not operatewhenthe firmssupplyservicesand the customersmayhaveto wait in
queue.
Supposethat only one firmreducesits price. Some of the customerswill now prefer
to receivethe serviceat a lowerprice,but this causesthe queueat the low-pricefirmto get
longeras an increasingnumberof customersturnto it. In the high-pricefirm,on the other
hand, the queuebecomesshorterand waitingtime is reduced. Customerswho sufficiently
prefera shorterwaitingtime will not go to the low-pricedfirm even though the price is
fromthe high-pricefirmto the
lower. At some stagethis processof customerstransferring
low-pricefirmwill come to an end. We show later that the point wherethe processends
can be an equilibrium.
Beforeturningto the detailedmodel, let us clarifythe differencebetweenour model
and a model of two firmssellingdifferentgoods. The waitingfor a serviceis clearlypart
of the serviceitself. If thereis a differentwaitingtimein eachfirm,thenthe serviceproduced
by eachfirmis a differentcommodity. But if the commodityis not the samein both firms,
then of coursethe priceneednot be the same. The mainfeatureof our problemis that we
cannotsay a prioriwhethertherewill be two commoditiesor only one: that is, whetheror
not the waitingtime will be the samein both firmsis determinedby the systemitself.
A similarproblemwas discussedby Smithies[6] in his paperaboutduopolisticspatial
competition. In both modelstherearetwo firms,two prices,andpartitionof the customers
betweenthe two firms. In Smithies'modelthe customeris affectedby freightrates,while
in our model the customerconsidersalso the waitingcost. The main differenceis that in
our modelthe priceis the only independentvariable(waitingtime is a dependentvariable),
whilein Smithies'modelboth priceandlocationareindependentvariables. Thisdifference,
519
REVIEWOF ECONOMICSTUDIES
520
and the fact thatwaitingtime is not the sameas freightcost, does not allowus to compare
the resultsof the two models.
But in one respecta similaritycan be found. We will show that usuallyboth firms
chargedifferentpricesand the streamsof customersare not the same, but if no customer
leaves without service-the prices are the same, and each firm serves one-half of the
customers. This last resultis equivalentto the case in Smithies'model,whereone firmis
located at each quartileof the marketand servesexactly one-halfof the customers,and
the pricesare the same. Smithiesshows the conditionsthat are necessaryfor this result.
But, undergeneralconditions,the firmsare not at each quartile,and the pricesmay differ.
Assumea systemwith a servicerenderedby two firms. In both firmsthe qualityof the
serviceitselfis the sameandthe servicetimeis distributedexponentiallywiththe samemean
service-timeparameter. Each firm fixes its serviceprice so as to maximizeprofit. We
assumethat thereis no co-operationbetweenthe firms. (Laterin the paperwe shall relax
this assumption.)
Thereis a Poissonstreamof customersarrivingat the firms. Eachcustomerhas three
alternatives-to choosefirm 1, to choose firm2, or to leavewithoutreceivingservice. The
choice mustbe madeby consideringthe cost of waiting,the expectedwaitingtime in each
firm,and the priceof the servicein each firm.
Twocomputercentreswhichcompetein the samemarketis an exampleof sucha model.
The expectedwaitingtime and the pricesin both centresareknownto all the customersand
potentialcustomers. A potential customeris includedin the streamof customerseven
thoughhe neverbothersto show up at eitherline becauseof the high pricesand the long
expectedwaiting time. By assumptionthe customer'schoice is made accordingto the
pricesandthe expectedwaitingtime and is not dependenton the currentwaitingline. That
is, we assumea high transactioncost of movingif the customeris alreadyat one of the
centres.
We assumethat the cost of waitingis a linearfunctionof waitingtime (here defined
as the sumof queuingtimeand servicetime)and thateachcustomerhas a differentwaitingcost function,so that each customeris characterized
by the parameterC whichdenotesthe
cost of waitingone unit of time. We assumethat thereexistsa distributionfunctionwhich
shows the probabilitythat the waitingcost parameterdoes not exceed C.
We shall use the followingnotation:
= the rewardfrom receivingthe service(assumedequal for all customers)
R
1/V
=
expected service time
Wi =-expectedwaitingtime at firmi (i = 1, 2)
C = waitingcost per unit of time (it is assumedthat C> 0)
Pi = the priceof the serviceat firmi (it is assumedthat P1 >P2)
Ui
= the net gain of a customer receiving the service from firm i
F(C) = the distributionof the C parameteramongcustomers(assumedcontinuous)
f(C) = the densityfunctioncorrespondingto F(C)
= the Poisson parameterof the flow of potential customers. For simplicitywe
assume A = I
Ai
= the Poisson parameter of arrivals at firm i.
The customer'snet gain at firmi is:
Ui =R-WjC-Pi
(i =1, 2)
....(1)
The customerwill preferfirm2 when U2> U1, that is, when (P1-P2)/(W2- W1)>C
(from R- W2C-P2 >1R- W1C-P1), and vice versa for firm 1. He will preferto leave
withoutservicewhen the net gain is negativein both firms,that is, accordingto (1), when
C>(R-Pi)/1W
for i = 1, 2.
LUSKI
A QUEUING SYSTEM WITH TWO SERVERS
521
Let us summarizethese conditions. The customerprefersfirm 2 when his waiting
cost satisfies
O<C ? (P1-P2)/(W2-W1)
... (2A)
and C < (R-P2)/W2.
He prefersfirm 1 when
...(3A)
and C?<(R-P1)/W1
C>(P1-P2)/(W2-W1)
and he leaves withoutservicewhen
and
C>(R-P1)/W1
... (4A)
C>(R-P2)/W2.
We assumethat both firmsare operatingand that at least one customerarrivesat each.
It follows that:
..*(5)
(Pl -P2)/(W2 -Wj) < (R - P&/W,
Underthis assumption,the above conditionsare reducedto the followingthreecases:
The customerprefersfirm2 when:
C _ (P1 -P2)/(W2-WD... (2)
He prefersfirm 1 when:
... (3)
(P1-P2)/(W2-W1)<C < (R-PP)/WW
and he leaveswithoutservicewhen:
.(4)
C>(R-PP/Wl.
When (2) holds (i.e. U2> U1) and accordingto (5), we have C< (R-PP)/WW which
implies U1> 0 and hence U2> U1>0. Thus (2) is a sufficient(and of course necessary)
conditionfor the customerto preferfirm2. However,when (4) holds (i.e. U1<0), then
from (5), (2) does not hold and U2< U1< 0. Thus (4) is a sufficient(and necessary)condition for the customerto leave withoutservice.
In this model, the choice of firmdependsnot only on the pricesin each firmbut also
on the expectedwaitingtime. But the expectedwaitingtime and the customer'sdecisions
are interdependent.
Given any price pair P*, P* the customer-flowto each firm will settle down to two
Poisson processeswith expectedwaitingtimes W1, W2*.We are concernedonly with the
expectedwaitingtime at equilibrium.
Let us now definethe functionsthat determinethe expectedwaitingtime. From the
literatureon queuingtheorywe can use the standardresultfor expectedwaitingtime in a
Poissonprocess: Wi = (V-2i) - (for examplesee Saaty[4, p. 342]). 1/ Vis the expected
servicetime and ti is the parameterof the Poisson streamof arrivalsat firm i. All the
customerswhose waiting cost satisfies(2) will prefer2, and the relevantfractionof the
customerstreamis exactly F[(P1 -P2)(W2 - W1)]. Accordingly,the parameterof the
Poisson streamat firm2 is
22 = F[(P1-P2)/(W2-W)]
(6)
(The right-handside is here multipliedby i, omittedbecausewe have assumedthat
= 1.) Accordingto (3) the fractionof the streamthat prefersfirm 1 is:
21= F[(R-P1)/W1] -F[(P1 -P2)/(W2-
Using WI= (VV
W)]..
)-f and putting
(P1-P2)(W2-
W1) =
W, =
Oc,(R-P&)/W1
we get the followingexpressionsfor the expectedwaitingtime in each firm:
W, = [V-F()
+ F(o))].
W2= [V-F(oe)]-1.
(
... (9)
522
REVIEWOF ECONOMICSTUDIES
Equations(8) and (9) definethe expectedwaitingtimein eachfirmas a functionof the
serviceprice.
We have so far assumedthatP1 >P2. But if the priceis the samein both firms,some
difficultiesarise. First, ocis not definedwhenP1 = P2 for it then follows that the expected
waitingtimes must be the same, i.e. W1 = W2,so that the denominatorof the expression
is zero. Second,in the precedingdiscussionwe could distinguishbetweenthe customersof
the two firms accordingto their waiting cost. With equal prices we cannot make this
distinction. Exactlyhalf of the total streamarrivesat each firm.
In orderto avoid these difficulties,we can look at the limitingvaluesof xcas Pl--P2.
Now W1must be close to W2as P1 approachesP2, and for the expectedwaitingtimes to
be similar,the arrivalratesof each streammustalso be similar. Thatis, 21-*2 as P1-+P2.
Using the oc,,Bnotation,the ti streamparametersare
21 = F(fJ)-F(a)
22 =
In the limit when 21-22,
F(a)
..
we have:
F(a)= IF(#).
This equationimplicitlygives the limit of acas P1-+P2.
..4.(10)
3. EQUAL OR UNEQUAL PRICES
Assume that the firms'costs are constant and independentof the stream of customers.
The profitfunctionof firm1 is 7C1 = P1)1 for 21 is the expectednumberof arrivalsat firm 1
per unit of time and hence,P1)1 is the expectedprofitof firm 1 per unit of time. Firm 1
assumesthat the serviceprice of firm 2 is constant, and chooses its optimumprice P1.
But 21 is also a functionof P1, and we formulatethe followingmaximumproblem,where
the constraintsdenote the relationshipbetween2) and P1:
max
P1, Wi, Ai
7C1 =
P1*(ll)
subjectto (6)-(9)and with P2 assumedconstant. Similarly,for firm2 we get
max 7r2 = P2
.(12)
P2, Wi, Ai
subjectto the sameconstraintsand with P1 assumedconstant.
Each firm chooses an optimumprice for every price of the other firm, and a pricechangingprocesstakesplacein the two firms. Theprocessends,withthefirms'equilibrium,
when neitherfirmwants to changeits own price. Let P*, P* be the priceschosen at this
point.
The main questionof this paper is whetherP* = P* is consistentwith equilibrium.
We analysethis questionas follows: first, the partialderivatives07t1/0P, and 07r2/OP2are
found; second, the values of these partial derivativesare calculatedat points where
P1 = P2. P = P* are equilibriumpricesif
?7t1/0P1?0 for all P1 > Pr
...(13)
07r2/OP2? 0 for all P2 < P2*
Theseare the generalconditionsfor equilibriumprices. In the specialcase wherethe
demandfunctionsare continuousand withoutkinks, the conditionsmust hold with strict
equality(see Shubik[5, p. 155]).
We will showthat the demandfunctionis not continuousat this point, so that the only
way to writethe conditionsfor a Nash equilibriumpoint is by inequalitieslike (13).
LUSKI
A QUEUING SYSTEM WITH TWO SERVERS
523
If a7t1/0P1>a7r2/aP2wheneverP1 = P2, then conditions (13) for equilibriumwith
equal pricesdo not hold and the systemis not in equilibriumwhen both firmssell at the
same price.
Let us begin with a7t2/0P2. From (12) and (6) we get the profitfunctionof firm2:
7r2 =
P2F(ca)
...(14)
whose partialderivativewith respectto P2 is
[
.(15)
a72 -F(o)_
P2f(o)
OW2 _ OWl
-w1OP2
Wl
OP2 DP2
In orderto obtainthe factor(0 W210P2 -0 W1/0P2) in termsof the parameters,differentiate (8) and (9) with respectto P2. Substitutionin (15) gives:
cL'
a 07t2
= F(x)-P2 [W2W1-+ W2c2 + ~~~W2
,
...(16)
v
OP2
f@a)
wherey = 1+ Wlf(f3)fl. If P2 = 0, then 07r2/OP2= F(oc)> 0.
As P2->P1 (whenP2 _ P1), W2->W1. Hence
_
L
-
lim
= F(x)-
W2a\c+cy/7
OP2
where W - W, = W2 (in the limit).
For firm 1, the derivationis similarand we get
P2P
lim
-__
-
F(x)_ P1 [f(3)Wax+1]Iy
w2
OP1
Pi
... (17)
...(18)
a + a/y
We can now look at the relationshipbetweenthe two derivatives(17) and (18). Using
y definedas above, the last derivativecan be written:
lim
1
and using (17),
-71
O2AP1
1
- F(o)_(-
lim
P1-P2
___
W2
a+ a/y
+ P1 f(/3)W(/3-ot)
W2
(vc+o)
=
OP1
OP2
W
a(y+1)
..4.(19)
... (20)
Now ,B>oc, hence the right-hand argument is positive. Therefore
0rl/OP1 >_ 7r2/OP2..(1
and the equalityholds only iff(,3) = 0.
= 0 and the secondWhen P* 0 P*, the two firmsare in equilibriumwhen arci/OPi
order conditionsare right. But when P* = P*, then, as shown earlier,the firms are in
equilibriumif conditions(13) are satisfied. But, as we have just seen, whenf(J3)>0, we
have inequalityin (21) for all Pi pairs whereP1 = P2; that is, conditions(13) are not
satisfied.
Now,f(,B)is the probabilitythat thereare customerswith high waitingcosts who want
to receivethe servicebut leavethe queuebecausetheyhaveto waittoo long. Thusf(J3)= 0
if all customersreceivethe service,while if some customersleave without being served,
f(O3)> 0, and the pricecannotbe the samein both firms. The difficultyis that,Bdependson
the price, so that we cannotsay whetheror not priceswill in fact be equal.
Assume,however,that the systemdiverges(thatis, A/2> V). Thenif all the customers
remainin the queue,the waitingtimewill extendto infinity,and,in such a system,f(13)is
positive. In otherwords,in a systemwhereA/2> V,each firmwill sell at a differentprice.
On the otherhand,if A/2is verysmallcomparedwith V,waitingtime will be relatively
short,even if all the customersreceivethe service. In that case, we have F(fl) = 1, so it is
reasonablethatf(fl) = 0 and that the pricewill be the samein both firms.
REVIEWOF ECONOMICSTUDIES
524
We can summarizethese resultsin the followinglemma:
Lemma. Suppose that the set on whichf(x) >0 is a single interval. Then, a necessary
conditionfor equalprice equilibriumP is:
R > W2AF-'(0o5) + WF- 1(1)3,
....(22)
where W is the expected waiting time in each firm such that each firm serves one-half of the
stream, and no one leaves withoutservice.
(Note that A = 1, but appearsherein orderto make it clearthat the partsof the right
hand have the same unit scale.)
Proof. If P1 = P2 = P, thenf(13)= 0. The conditionfor maximumprofitin firm 1
becomes(see 19):
07pl -
F(x)-2
(2
)
=
... (23)
From the fact that F(a) = A/2and A = 1 we obtain:
...(24)
P= AW2c =,AW2F-'(0-5)
f(,B)is zero if
(R-P)/W > F-1(1).
By replacingof P from (24) we obtainthe necessarycondition(22).
The meaningof the conditionfor equalprices(22) is that no customerleaveswithout
service. That happenswhen even for the highestwaitingcost per hour, we have that the
benefitfrom the serviceR is greaterthan the sum of the total waitingcost [WF-'(1)] and
the pricefor the service[W2AF- '(0.5)].
Unfortunately,condition(22) is not sufficientfor equal price equilibrium(a counter
examplecan be found). The reasonis that we cannotbe surethatf(,B)has vanished. The
firmsmay end up with such high pricesthat some customersleave withoutservice.
4. CO-OPERATIONBETWEENFIRMS
In this section we relax the assumptionof no co-operation. The two firmsnow aim at
maximizingthe sum of theirprofits,that is, max Xr= ir1+X2. We considerhere whether
theirpriceswill be equal. We firstdeterminethe necessaryconditionsfor maximumprofits
underequalpricesand then calculatehow profitschangewhen one priceis changed.
We start by setting P, = P2 = P, that is, by forcing the prices to be equal. The
intensityof the streamof customerswho do not leave withoutserviceis A:
A=
... (25)
F[(R-P)/W)].
Total profitis i = )P. Each firmreceiveshalf of the total streamof customers,and
expectedwaitingtime in each firmis therefore
W = (V-212)`
Using (25) and putting (R-P)/W
= ,B,we can write
max ir= PF()
... (26)
p
s.t.
W
[V--'F(f3)]'
with solution:
I,BF(f)+
(=
Wf(fl)
W2
0.
... (27)
LUSKI
525
A QUEUING SYSTEMWITH TWO SERVERS
The P* whichsatisfies(27) maximizesthe joint profitof the two firms,underthe constraintof equalprices.
Let us now allow the priceof one of the firmto change. In orderto demonstratethe
effecton joint profitswe have to determinethe sign of the followingexpression:
ap1
=
2
+
ap1
...(28)
a
wherethe derivativesarevaluedat the pointsP* = P* = P* whichsatisfy(27). Thesolution
for the left-handtermis equation(18), while the right-handtermis given by
P2 1- [W1P2cf(f)]/y.
0t2_
...(29)
W2
ap
c+ c/y
Substituting(29) and (18) in (28) yields:
__
OP,
=
F(L)_
P1 [Waf()+1]/y
a +i/y
W2
+ P2 1-[WP2cAf()]y
W2
a + ccy
.
(30)
We can also obtain ai/IP1 in the form:
F(f3)
f3#F(f3) P(213'31
A anr
c2
(Pa Wf(1) +
...(31)
whereA is positive.
We are looking for the sign of the derivativeai/IP1 when it is valued at the point
=
P* P* = P* the optimumprice for the two firmsunderthe constraintof equal service
prices. That is, P* must satisfy(27). Then subtracting(27) from (31) gives:
P
A an = _ w
ap,
l-
>0
...(32)
because,B> a).
is always positive and the two firms can always increasetheir total
Thus, 07r/OP1
profitby sellingtheir servicesat differentprices.
That is an obvious result following the assumptionthat differentcustomershave
differentwaiting costs. In the initial case, the two firms chargethe same price. Then,
one firm can raise its price without any changein the total numberof customers. The
resultis higherjoint profits. Supposethat the prices are the same in each firm, and the
customersare dividedbetweenthe firmsso that those with higherwaitingcost are served
in firm 1. Now, let us take the one with the lowestwaitingcost in firm 1 and transferhim
to the secondfirm. A decreasein the expectedwaitingtime in firm 1 will enablethe firm
to raiseits pricein such a way that those who were servedin firm 1 continueto be served
there. In conclusion,one firmraisesits price,the secondfirmdoesn'tchangeits price,the
total numberof customersremainsthe same, and the resultis higherjoint profits.
First versionreceivedFebruary1973; final versionacceptedFebruary1976; (Eds.).
This paper is based on a Ph.D. dissertationsubmittedto the Hebrew University of Jerusalem. I am
indebtedto ProfessorsM. E. Yaari, D. Levhariand E. Sharonfor many commentsand suggestions,and to
the editors and an anonymousrefereefor useful commentson earlierversions of this paper.
REFERENCES
[1] Edelson, N. M. " CongestionTolls Under Monopoly ", AmericanEconomicReview(December1971),
873-882.
[2] Knudsen, N. C. " Individualversus Social Optimizationin QueuingSystems", Econometrica(1972),
515-528.
[3] Naor, P. " The Regulation of Queue Size by Levying Tolls ", Econometrica(January1969), 15-23.
[4] Saaty, T. L. MathematicalMethodsof OperationResearch(McGraw-HillBook Company, 1959).
[5] Shubik, M. Strategyand Market Structure(John Wiley & Sons, Inc. 1959).
[6] Smithies,A. " OptimumLocation in Spatial Competition", Journalof Political Economy(February
1941), 423-439.