modeling and managing the percentage of satisfied customers in hidden and revealed waiting line...

Modeling and Managing the Percentage ofSatisfied Customers in Hidden and Revealed

Waiting Line Systems

Chester Chambers • Panagiotis KouvelisCox School of Business, Southern Methodist University, Dallas, Texas 75275-0333, USA

School of Business, Washington University, St. Louis, Missouri 63130-4899, [email protected] • [email protected]

We perform an analysis of various queueing systems with an emphasis on estimating a singleperformance metric. This metric is defined to be the percentage of customers whose actual waiting

time was less than their individual waiting time threshold. We label this metric the Percentage ofSatisfied Customers (PSC.) This threshold is a reflection of the customers’ expectation of a reasonablewaiting time in the system given its current state. Cases in which no system state information is availableto the customer are referred to as “hidden queues.” For such systems, the waiting time threshold isindependent of the length of the waiting line, and it is randomly drawn from a distribution of thresholdvalues for the customer population. The literature generally assumes that such thresholds are exponen-tially distributed. For these cases, we derive closed form expressions for our performance metric for avariety of possible service time distributions. We also relax this assumption for cases where service timesare exponential and derive closed form results for a large class of threshold distributions. We analyzesuch queues for both single and multi-server systems. We refer to cases in which customers may observethe length of the line as “revealed” queues.“ We perform a parallel analysis for both single andmulti-server revealed queues. The chief distinction is that for these cases, customers may developthreshold values that are dependent upon the number of customers in the system upon their arrival. Thenew perspective this paper brings to the modeling of the performance of waiting line systems allows usto rethink and suggest ways to enhance the effectiveness of various managerial options for improvingthe service quality and customer satisfaction of waiting line systems. We conclude with many usefulinsights on ways to improve customer satisfaction in waiting line situations that follow directly from ouranalysis.

Key words: waiting lines; customer satisfaction; management of customer queuesSubmissions and Acceptance: Received December 2003; revision received December 2004, August 2005,

October 2005; accepted December 2005 by Kalyan Singhal.

1. IntroductionMany settings exist in which customers join a single-channel queue with a high level of commitment toreaching the service provider at the end of the line. Forexample, we may join a queue to check in at an airportwith tickets in hand, or to receive a critical medicaltreatment, or to handle a particularly pressing per-sonal or business matter. In such settings, each cus-tomer brings with him/her some tolerance for thewait that is to take place. If customers are polled afterthe experience and asked questions such as, ‘Was thewaiting time excessive?’, some customers will answer

in the affirmative while others answer in the negativeeven if they all experience waits of identical lengths. Inthis work, we will assume that each customer bringswith him/her some threshold value (X) such that, aslong as the actual wait (W) is less than this value (W� X � 0), he would respond that the wait was notexcessive. For ease of exposition, we label all custom-ers with W – X � 0 as satisfied and all customers withW � X � 0 as dissatisfied. We label the portion ofcustomers for which W � X � 0 as the Percentage ofSatisfied Customers (PSC.)

A large body of literature focuses upon objective

POMSPRODUCTION AND OPERATIONS MANAGEMENTVol. 15, No. 1, Spring 2006, pp. 103–116issn 1059-1478 � 06 � 1501 � 103$1.25 © 2006 Production and Operations Management Society

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measurements such as average waiting time or thelikelihood of experiencing a wait. Such measurementsare most useful when compared to some benchmark.Our approach addresses this by comparing the wait-ing time to customer specific thresholds. While ourbinary classification scheme is a dramatic simplifica-tion of reality, our approach remains valuable for sev-eral reasons. First, our metric provides a simple, globalmeasurement of system performance that can easilybe conveyed, and explained. Second, our metric gen-erates values that may easily be verified in practice.Data from exit surveys or other customer responsesallow us to classify customers as satisfied/not satis-fied, while estimating the ‘degree of dissatisfaction’ asa function of the waiting time introduces measure-ment difficulties that may be impossible to overcome.Third, the analytical statement of our metric allowsmanagers to quickly estimate the impact of proposedsystem changes.

1.1. Waiting ThresholdsThe notion of waiting time thresholds is common inthe literature on call centers (Whitt 1999a,b). In thatsetting, this value is interpreted as the time afterwhich a customer will hang up after being put onhold. This behavior is often labeled ‘reneging.’ In ourmodels, we typically assume that there is some char-acteristic of the service that minimizes this behavior.(The appendix does include limited results which in-clude balking and reneging for M/M/c queues.) Ournotion of a waiting time threshold is better describedas the length of time a customer will wait, prior to theinitiation of service, before becoming dissatisfied withthe service provider. Our focus on the time prior to theinitiation of service is consistent with the arguments ofMaister (1985) who suggested that these ”pre-process“waits are often (but not always) more unpleasant thanin-process waits. This argument is further supportedby empirical evidence (see Dube-Rioux, Schmitt, andLeClerc 1988 and the references therein.)

In many cases, customer satisfaction (or dissatisfac-tion) relates to the gap between customers’ expecta-tions of the performance of the service system and itsactual performance (see Maister 1985 and Boulding etal. 1993 for empirical support.) This implies that cus-tomer threshold values are likely to be correlated withtheir expectations. This suggests that management ofthese expectations should have a significant impact onthe PSC. On the other hand, it is also clear that thesethreshold values are not fully determined by customerexpectations. One may be dissatisfied with a wait evenwhen it is exactly as long as was expected.

1.2. Selected LiteratureMost of the research on queueing has dealt with themathematical theory of waiting lines, and descriptions

of waiting time distributions have been developed forvirtually any setting. (For recent textbooks, see Wolff1989; Hall 1991; Gross and Harris 1998.) Several re-searchers have expanded this large body of work byfocusing upon customer satisfaction measures whichseek to include customer-specific attributes in theircalculation. (For examples, see Carmon et al. 1993;Green and Kolesar 1987, 1988; Whitt 1992a,b.)

The work of Whitt (1999a,b) has been particularlyinfluential in this area. These papers focus upon callcenters, and all of the closed form metrics derivedassume exponentially distributed inter-arrival times,service times, and threshold values. It is useful to havecompact expressions for the PSC for a number ofsettings where we may relax at least one of theseassumptions. For example, the assumption of expo-nential service times is not universally applicable evenfor call centers (see Inman 1999). For data on non-exponential service times in call centers, see Wardell etal. 2001. Our analysis will consider other service timedistributions in such a way that we still obtain closedform expressions for the PSC.

The work of Whitt is also restricted to the consider-ation of cases in which the customer thresholds areindependent of the state of the system. This is quitenatural for call centers. We refer to such settings as“hidden queues” because some knowledge of the sys-tem state is hidden from the customer’s informationset. However, many cases exist in which the customercan easily see the number of customers ahead of him/her awaiting service. We refer to these settings as“revealed queues.”

Common examples of hidden queues includeamusement parks where the length of the line is hid-den by bending it around corners or using stagingareas, medical settings where some people in a wait-ing area are accompanying customers and some cus-tomers hold appointment slots even if they are notpresent, and many electronic systems in which cus-tomers cannot see how many jobs are ahead of them inline. Common waiting line situations that qualify asrevealed queues include check-out areas in supermar-kets, lines to enter unique sporting or entertainmentevents, lines at licensing centers, bus stops, theaters,etc.

2. Hidden and Revealed M/G/1Queues

Let us consider a single-line queueing system with asingle server. Customers arrive according to a Poissonprocess, and the required service time follows a gen-eral distribution function. Upon arrival each customercan be characterized as holding some threshold valuefor waiting in the line. Each customer’s threshold willdepend on a variety of unknown, customer-specific

Chambers and Kouvelis: Modeling and Managing the Percentage of Satisfied Customers in Hidden and Revealed Waiting Line Systems104 Production and Operations Management 15(1), pp. 103–116, © 2006 Production and Operations Management Society

factors. We model this by having each customer ran-domly drawing a threshold value out of a prespecifieddistribution. We assume that our queue is part of astationary M/G/1 system. Our notation is rather stan-dard, and is summarized in Table 1.

2.1. Hidden M/G/1 Queue With ExponentialDistribution of Customer Thresholds

We label the random variable of interest as Z, definedas W-X. We want to calculate the distribution functionof Z, let us call it F, and in particular F(0), whichrepresents the probability that a randomly selectedcustomer is satisfied. Since we assume Poisson arriv-als, F(0) is also the PSC. Since the randomness in Xarises from the heterogeneity of the customer popula-tion, we assume that X and W are statistically inde-pendent when performing our calculations. If the cus-tomer’s expectation, X is drawn from an exponentialdistribution with rate �, then the quantity of maininterest can be expressed with the use of the followinglemma.

Lemma 1. Let I and J be independent non-negativerandom variables. I has a distribution function GI and J isexponentially distributed with rate �. Then P{I � J � 0}� �(�), where � is the Laplace-Stieltjes (L-S) transform ofGI.

See Appendix for all proofs and additional mathe-matical details, including comments on numerical ap-proaches for more general cases. Using Lemma 1, wehave,

F�0� � P�Z � 0� � P�W � X � 0� � H�� , (1)

where H is the L-S transform of H(x) � P{W � x}. For

the stationary M/G/1 queueing model, it is known(see Kleinrock 1975) that

H�s� � �0

�

e�sxdH� x� �s�1 � ��

s � � � �B�s�, (2)

where B(s) is the L-S transform of the service timedistribution function. Thus, from (1) and (2), we obtain

F�0� �� 1 � ��

� � � � �B�� . (3)

When X Exp(�), we can use Equation (3) to state F(0)for cases where the L-S transform of the service timedistribution is known, and can be evaluated at �. Suchexpressions for a variety of service time distributionsare shown in Table 2.

A variety of service time distributions are consid-ered because delivery systems differ so widely in theirstructure. While exponential or Erlang service timedistributions are quite common in the literature, auto-mation or process standardization often leads to ser-vice times that are virtually deterministic. Hyperexpo-nential service times may be present when differentclasses of customers need different levels of service,but the provider has no way of knowing which class acustomer is in until he/she reaches the service pro-vider. A Generalized Exponential service time ispresent when some portion of the population () canbe shifted to an alternate service provider or requireservice times of virtually 0. Finally, the ability to

Table 2 Percentage of Satisfied Customers in Hidden Queues When X Exp(�) and Service Times Follow Various Distributions

Distribution of service times Percentage of satisfied customers (F (0))

Exponential () ��1 � ��

� � � � �

� �

Gamma (�, �) ��1 � ��

� � � � � ��1

� � ��1��

Deterministic ��1 � ��

� � � ��

�

.

Erlang (k) ��1 � ��

� � � � �� k

� � k�k

Hyperexponential ��1 � ��

� � � � � �i�1

m

i� i

� � i�

Generalized exponential ��1 � ��

� � � � ��

� � �

Table 1 Notation Used Throughout This Work

� average arrival rate of customers per unit timeS service time random variableB service time distribution function (i.e. B(x) � P {S � x })Mi i-th moment of the service time (i.e., mi � E [S i ]) average service rate ( � 1/E [S ])� utilization factor (i.e., � � � E [S ])W stationary waiting time random variableh (generalized) stationary waiting time densityH stationary waiting time distribution function (i.e., H(x) � P {W � x })L number of customers in the systemX customer’s waiting time threshold upon arrival to the systemG distribution function of X (i.e., G(x) � P {X � x })n index for number of customers in the systemQn probability that there are n customers in the system (i.e., qn � P {L

� n})Hn waiting time distribution function given that there are n customers aheadYn customer’s expectation of the waiting time when (s)he observes n

customers in the systemGn distribution function of Yn

mn Kronecker’s delta (i.e., mn � 1 for m � n, 0 otherwise)U(x) step function defined by U(x) � 1 if x � 0 and U(x) � 0 for x � 0.

Chambers and Kouvelis: Modeling and Managing the Percentage of Satisfied Customers in Hidden and Revealed Waiting Line SystemsProduction and Operations Management 15(1), pp. 103–116, © 2006 Production and Operations Management Society 105

model Gamma distributed service times is useful be-cause it is a good fit for a wide range of settings notsatisfactorily captured by simpler distributions.

2.2. Hidden M/M/1 Queue With GeneralDistribution of Customer Thresholds

While it is common to assume that X Exp(�), Lemma1 also allows us to deduce closed form expressions forF(0) when service times are assumed to be exponentialas long as the L-S transform of the distribution fromwhich customer threshold values are drawn is known.For this setting, Lemma 1 implies that we may expressthe percentage of satisfied customers as,

F�0� � �1 � �� 1 � G� � ��, (4)

where each customer’s threshold is drawn from adistribution function G, and G(s) is its L-S transform.Closed form expressions for F(0) for several such casesare summarized in Table 3.

If customers are promised a’priori that the waitingtime will be some specific value, say W0, then, virtu-ally all customers will become upset if the wait islonger than W0, and the distribution of X approachesthat of a deterministic setting. The notion of hyperex-ponential thresholds is applicable when considering amixture of classes of customers interspersed in thesame line. This is also useful in modeling a setting inwhich a portion of the population (I) is ”impatient“such that any wait is considered intolerable. A uniqueapplication of the Generalized Exponential distribu-tion exists when some fraction of the customer poolmay be distracted, entertained, or treated in such away that their perceived cost of waiting is 0. Increas-

ing this fraction (p) of ”patient“ customers obviouslyleads to increases in customer satisfaction.

2.3. Revealed M/G/1 QueueIn this setting the arriving customer observes thelength of the waiting line prior to the point at whichhis/her threshold value becomes fixed. The thresholdsetting process is modeled as a draw from a pre-specified distribution, the parameters of which de-pend on the length of the line that the customer ob-serves upon arrival. The manager of such a servicedelivery system is concerned with the PSC over allpossible states of the system, and the unconditionaldistribution F(x) is of primary importance. This distri-bution may be expressed as,

F� x� � �n�0

n��

qn Fn � x� (5)

In other words, the PSC must be defined by calculat-ing the PSC given that n people are in the system (Fn)as well as the probability that n people are presentupon customer arrival (qn.) We will deal with specialcases applying Lemmas 1 and 2 which yield closedform results. (Other approaches to this problem formore general cases are discussed in the appendix.)

2.3.1. The Revealed M/G/1 Model When Yn

Exp(�). If the distribution of customer thresholds isexponential for all states of the system, i.e., Yn Exp(�n) for all n, we can obtain closed form expres-sions for Fn(0) and F(0). Using Lemma 1, we concludethat

Fn �0� � Hn ��n �, (6)

where Hn is the L-S transform of Hn and is given by

Hn �s� � Bn�1�s�R�s� � Bn�1�s�1 � B�s�

m1 s . (7)

From (6) and (7) we have

Fn �0� � B�n�1��n �1 � B��n �

m1�n. (8)

For this special case, it also holds that

F�0� � �n�0

�

qn Fn �0� � �n�0

�

qnHn ��n �

� �n�0

�

qnB�n�1��n �1 � B��n �

m1�n. (9)

Table 4 summarizes Fn(0) values for a variety of ser-vice time distributions derived using (8). A variety of

Table 3 Percentage of Satisfied Customers in Hidden Queues WhenService Times Exp(�) and Customer Thresholds X FollowVarious Distributions

Distribution ofcustomer thresholds

Percentage ofsatisfied customers F(0)

Exponential () �1 � �� 1 � � �

� � � ��Gamma (�, �) �1 � �� 1 � � �

��1 � � � ��

Deterministic �1 � �� 1 �1

�� Erlang (k) �1 � �� 1 � � k�

� � � k��k�

Hyperexponential �1 � �� 1 � �i�1

m

i� �i

� � � �i��

Generalizedexponential

�1 � �� 1 � �

� � � ��


sources may be used to estimate qn values for M/G/1queues such as the tabulated results included in manytextbooks. Thus, we can calculate Fn(0) directly formany settings making it easy to calculate F(0).

2.4. The Revealed M/M/1 Model for VariousDistributions of Yn

With exponential service times we have, R(s) � B(s)and Hn(s) � Bn(s) . For the special case whereYnExp(�n), we have

Fn �0� � �

� �n� n

, and (10)

F�0� � �n�0

�

�1 � ��nFn �0� � �1 � �� n�0

� � �

� �n� n

.

(11)

For cases in which Yn is not exponentially distributed,we may make use of the fact that the waiting time isthe sum of n exponential variables and solve the fol-lowing equation

P�W � X� � 0� � 1 � P�X � W� � 0� (12)

� 1 � �0

�

Fx �w� g�w��w

where g(w) is the p.d.f. of an Erlang distribution withparameters k and . Fx(w) is the c.d.f. of the thresholdX, evaluated at w. If we set � � k, we may state thisquantity as

Fn �0� � 1 ��n

�n � 1�! ��1�n�nF��

��n . (13)

In other words, if we know the nth derivative of theL-S transform of the threshold distribution, we cancalculate Fn(0). We then combine these values with qnvalues to calculate F(0).

2.5. M/M/1 Queues With ExponentiallyDistributed Thresholds, Balking, andReneging Customers

Our analysis may be extended to include both balkingand reneging behavior among customers. For the spe-cial case in which we have an M/M/1 system wherethe probability of a customer balking can be stated asa function of the length of the line (1 - bn), we knowthat the percentage of customers who balk is simply, �� ¥n�1

� qn(1 � bn). We label the rate at which custom-ers renege when there are k customers in front of themas k � ¥j�1

k j. If we label the percentage of arrivingcustomers that do not balk or renege and are satisfiedas F�(0), we can now state that F�(0) � ¥n�0

� qnbn( j/ 2 j)

n Fn(0). (See appendix for additionaldetails.) We label the percentage of reneging custom-ers as � and state the value along with the PSC as

� � �n�1

�

bn qn�1 � Wn � ��, and (14)

F�0� � F��0��1 � � � ��. (15)

3. Observations on Hidden andRevealed M/G/1 Queues

For the case of hidden queues, we assume X Exp(�).For the case of revealed queues we assume thatYnExp(�/n�), where 0 � � � 1. We focus upon thisfunctional form because it appears reasonable to assumethat Yn should be both increasing and concave in n.When � � 0 threshold levels are independent of n. Weassume that � � 1, since it seems unreasonable to as-sume that customer thresholds would grow faster thann. Considering the scenarios discussed in Tables 2 and 4we derive several useful results including the following.

Proposition 1. For hidden queues when X Exp(�),or revealed queues when Yn Exp(�/n�) the percentage ofsatisfied customers F(0) is:

(i) a non-increasing, convex function of � for the M/G/1model;

(ii) a non-increasing, concave function of � for theM/G/1 model;

(iii) a non-decreasing, concave function of k for theM/Ek/1 model; and

(iv) a non-decreasing function of (1 - ) for theM/GE/1 model.

Table 4 Percentage of Satisfied Customers in Revealed Queues Whenn Customers are in the System, Customer Thresholds Yn

Exp(�n) and Service Times Follow Various Distributions

Distribution ofservice times Percentages of satisfied customers (Fn(0))

Exponential (1/) �

�n � �n

Gamma (�, �) 1��n

� ��

�� n��n�1 � � ��

�� n��

Deterministic � 1�n

�n��n � 1�n

�Erlang (k) k

�n� k

�n � k�k�n�1��1 � � k

�n � k�k�

Hyperexponential1

�n �i�1

mi

i

�i �i�1

mi

�n � i�n�1�1 � i �

i�1

mi

�n � i�

Generalizedexponential

�n�

�n � �n�1�1 �

�n � �


Proposition 1 implies that when Yn Exp(�/n�), ashift in the mean of the distribution from which cus-tomers’ expectations are drawn has a relatively sizableimpact on PSC. This is particularly important when �is low (part i) or when � is high (part ii). By increasing kfrom 1 (the exponential case) toward infinity (the deter-ministic case), we can display the impact that reducingservice time variability has on the PSC. For example,consider the setting in which � � 0, � 100, � isextremely small (0.00001), and � is extremely high (�� 99.9%). Reducing service time variability has thegreatest impact in the most extreme cases, such as thisone. For this instance, changing k from 1 to infinityincreases PSC by 16.9% (from 41.7% to 58.6%). Alterna-tively, an equivalent increase in PSC could have beenobtained by dropping � to 99.5%. In this sense, we seethat changing � or � has a far greater impact on PSC thanaltering k. Figure 1 shows PSC values as a function of kwhen � is fixed at 0.7 for various levels of �. We note thatchanging � may produce a significant increase in PSC forany value of k and that the impact of increasing k ispositive but marginal by comparison.

To understand the effects of reducing the variability ofthe distribution of customer thresholds, we focus uponthe case in which expectations are drawn from an Erlangdistribution with parameter k. Consideration of our ex-pression for PSC leads to the following result.

Proposition 2. For the M/M/1 model, the percentageof satisfied customers F(0) is:

(i) a non-decreasing, concave function of k, when X (orYn) Ek

(ii) a decreasing, linear function of i when X Hm(i, �)(iii) an increasing, linear function of (1 – p), when X

GE(p, �).

This result is graphically depicted in Figures 2 and3. From Figure 2, we observe that the effect of areduction in the variability of customer thresholdsmay be realistically labeled as marginal. For this ex-ample, a reduction from Cx

2 � 1.0 to 0.1 leads to anincrease in the percentage of satisfied customers of0.018. Figure 3 depicts results noted in parts ii and iii.Part ii relates to a setting where some portion of thecustomer base (I) would best be described as impa-tient, meaning that any wait produces dissatisfaction.Part iii represents a setting in which the satisfaction ofa fraction of customers (1 � p) is independent of thewaiting time. The other customers have threshold val-ues drawn from an exponential distribution withmean �. In this case, the PSC rises linearly with (1� p), when � is held constant. A unique line exists foreach � value on Figure 3, and its slope rises as � rises.These two sets of curves are shown together to em-phasize the finding that when all else is equal, growthin the ”patient“ segment of the population is moresignificant than an equal growth in the ”impatient“segment of the population.

Figure 4 presents the result of an analysis of anM/M/1 system where customers balk upon finding asystem containing n customers with probability (1� bn) and renege at a rate . The results shown herecompare the PSC values under several scenarios. Case1 assumes no balking or reneging. In case 2, we con-

Figure 1 F (0) Vs. k for M/Er(k)/1 System With � � 0.70 and � � � � 100 at Various Levels of �.


sider balking at a rate of 5%. This value was suggestedby research on call centers (Salzman and Merotha2001) which found that roughly 5% of customers balkupon being put on hold even if the average waitingtimes are very short. In case 3, we consider renegingwithout balking, where �10, and in case 4, we treatboth forms of customer loss. The results shown inFigure 4 indicate that balking and reneging serve toincrease PSC values relative to the base case as arrivalrates rise. Balking reduces the effective arrival rate,and reneging makes the line move faster for the cus-tomers that chose to endure the wait.

3.1. Hidden vs. Revealed QueuesConsideration of our metric for M/M/1 queues leadsto the following result.

Proposition 3. For revealed M/M/1 systems with Yn

Exp(�/n�), and 0 � � � 1 the percentage of satisfiedcustomers Fn(0) is a monotone decreasing function of n.

It is intuitive to expect that longer lines will producelonger waits, reducing customer satisfaction. Proposi-tion 3 adds to this insight by showing that if customerthresholds grow no faster than the length of the wait-

Figure 2 F (0) Vs. k for M/M/1 System With X Er(k), � � � � 100, and � � 0.70.

Figure 3 F (0) for M/M/1 System With X H or X GE and � � � � 100 at Various Levels of �.


ing line, then satisfaction rates decrease as the linelength grows. Further analysis of Fn(0) also leads tothe following result.

Proposition 4. For revealed M/M/1 systems with Yn

Exp(�/n�), 0 � � � 1 the percentage of satisfied cus-tomers F(0) is a monotone increasing function of � and adecreasing function of �.

Figure 5 illustrates this result. It plots F(0) values asa function of utilization rates for the M/M/1 queuewhen � � 0 or 1. It is apparent from the figure that thegains from using a revealed queue grow with utiliza-

tion up to a point where � values are very high. (In thiscase, � � 0.95.)

4. Hidden and Revealed M/M/cQueues

4.1. Hidden M/M/c QueueConsider a single line M � M system with c servers.When customer thresholds are drawn out of an expo-nential distribution with parameter �, we can expressthe PSC as follows:

Figure 4 F (0) Vs. � for M/M/1 System With � � � � 100, Balking, and Reneging.

Figure 5 F (0) Vs. � for M/M/1 System With � � � � 100 at Various Levels of �.


F�0� � q0 �n�0

c�1 �c��n

n! � q0

��c�c

�c � 1�!

c � � � �, where

(16)

q0 � �n�0

c�1 (c�)n

n! � � (c�)c

c! �� 11 � ��1

. (17)

(Additional details are available in the Appendix.)

4.2. Revealed M/M/c QueueWhen customer thresholds are drawn out of an expo-nential distribution with its parameter �n, where n isthe length of the waiting line upon the customer ar-rival, the conditional waiting time distribution can beexpressed as,

Hn � x� � P�W � x�L � n�

� � c(c x)n�c

(n � c)! e�c x if n � c

U(x) if n � c(18)

where U(x) is the step function defined in Table 1.Using Lemma 1, we can easily calculate

Fn �0� � � � c

c � �n� n�c 1

if n � c

1 if n � c. (19)

Thus, when � � (�/c), the percentage of satisfiedcustomers is

F�0� � �n�0

�

qn Fn �0� � �n�0

c�1

qn

� �n�c

�

qn� c

c � �n� n�c 1

, and

F�0� � q0 �n�0

c�1 ��c�n

n! � q0 �n�c

��ncc

c! � c

c � �n� n�c 1

.

(20)

4.3. Revealed M/M/c Queue With Balking andReneging Customers

Analysis of the multi-server system with balking andreneging customers is surprisingly similar to that of asingle server system because the transform of thewaiting time distribution has a very similar structure(see Whitt 1999a). Specifically, we may state the L-Stransform of the waiting time as,

Fn �0� � Wn �s� � �j�0

k � c � j

c � j � s� (21)

In these equations, k indexes the position in the lineand c is the number of servers; therefore, n � c k.We also note that the q values are modified to reflecta multi-server system as follows,

qn � q0�n �

i�1

n bi�1

nfor 0 � n � c, qn

� q0�n �

i�1

c�1 bi�1

n�

c

� bi�1

c � , for c � n � �, and q0

� �1 � �n�1

c�1

�n �i�1

n bi�1

n� �

n�c

�

�n �i�1

c�1 bi�1

n�

c

� bi�1

c � ��1

.

(22)

4.4. Observations on Hidden and RevealedM/M/c Queues

For both hidden and revealed multi-server systems,the following result may be stated.

Proposition 5. For both the hidden and revealed M/M/cmodels, the percentage of satisfied customers, F(0) is:

(i) a non-increasing, concave function of � (assuming �or is fixed);

(ii) a non-increasing, convex function of �.

Figures 6 and 7 are illustrative of this result forhidden M/M/c queues. Figure 6 shows that for a fixedtotal system capacity (i.e. c � a constant), the PSCincreases as the number of servers increases. Thisstrongly implies that designing a system with more,but slower servers leads to higher levels of customersatisfaction. Figure 7 shows that the PSC tends tobecome less sensitive to a shifting of the mean of thecustomers’ threshold distribution as the number ofslow servers is increased. This added benefit furtheradvocates the use of a multi-”slow“-server systemover a single ”fast“ server queue.

In Table 5, we report the percentage of satisfiedcustomers for a hidden M/M/c model with X Exp(�) and for a revealed M/M/c model with Yn Exp(�/n).

The table reports results for different levels of utiliza-tion and for different numbers of servers c (c is constantfor all cases). From the results shown, we can infer thatrevealed M/M/c queues tend to dominate M/M/1queues in terms of the PSC. This result is entirely rea-sonable given the observation that while the distributionof the number of customers in the system is stochasti-cally smaller in the single-server case, the reverse holdsfor the number of customers in the queue (see Wolff1989, p. 258). In other words, visitors to the revealedsingle server queue are likely to see longer lines, and thePSC decreases as a revealed queue gets longer.


A result analogous to Proposition 4 can be stated forrevealed M/M/c systems with Yn Exp (�/n).

Proposition 7. For revealed M/M/c systems with Yn

Exp(�/n), the percentage of satisfied customers, Fn(0) isa monotone decreasing function of n.

The proof is omitted for brevity, but follows exactlythe same steps as the proof of Proposition 4.

5. Conclusions and ManagerialInsights

Our analysis and stated propositions lead to severalinsights that should prove useful for the managementof waiting line systems.

Insight 1. Reducing the psychological cost of waitingpromises great payoffs by increasing the percentage of sat-isfied customers.

Figure 6 F (0) Vs. � for M/M/c System With � � 0.70 and Various Levels of c.

Figure 7 F (0) Vs. c for M/M/c System at � � 0.70.


If the service provider can alter customers’ thresh-olds, then the firm is likely to produce a significantincrease in the number of satisfied customers. Ourresults also show that increasing the portion of thepopulation who don’t mind waiting significantly in-creases values of F(0). The distribution of thresholdvalues may be affected by a variety of means includ-ing providing information to reduce uncertainty, en-tertaining customers, making them more comfortable,or creating a greater sense of value for the end prod-uct.

Insight 2. For many systems, reductions in the vari-ability of service times or in the variability of customerthresholds have a lower than anticipated impact on the PSC.

This is not an argument that reduction in servicetime variability is unimportant. However, our resultsdo imply that reducing variability is less significantthan intuition might suggest. Extremely impatientcustomers are not likely to be satisfied with any no-ticeable wait, and extremely patient customers areeasy to satisfy. Our analysis suggests that the greatestimpact arises from repositioning the threshold valueof the ”average“ customer.

Insight 3. An analytical estimation metric for PSC iscritical for the management of many service delivery sys-tems.

As mentioned in prior research, including Kleinfeld(1988) managers routinely hear requests and advice tospend more money and offer more service. Our ap-proach presents an objective and measurable basis fordecisions regarding such policies. It is a simple matterto review survey data before and after a change ismade to access its impact, but it is a far differentmatter to offer rationally developed predictions about

satisfaction levels prior to a commitment to make suchchanges. Our metric provides one approach to helpevaluate alternatives.

Insight 4. Revealing information on waiting linelengths may be beneficial for service systems if it alterscustomer thresholds and lines are not likely to become verylong.

The impact of revealing queue length may be sig-nificant if customers alter their thresholds in a mannerfavorable to the firm. For cases in which � is low, suchshifts are of little consequence. On the other hand, if �is moderately high; say between 60 and 85%, thenpositive � values can provide a meaningful increase inF(0). For systems with very high � values; say over85%, the impact of customers raising their thresholdvalues in response to viewing the length of the line isoverwhelmed by the fact that the waits are very long.Another impact of providing information about thelength of the queue is that some customers who wouldhave been dissatisfied with the wait may chose to balkinstead, perhaps to return during off-peak times. Ifthis occurs, then the total percentage of customerswho are satisfied actually rises.

Insight 5. Assuming the same total system capacity,multiple slow server configurations have higher proportionsof satisfied customers than do configurations with a singlefast server.

Significant improvements in customer satisfactionoccur when systems move from a single ”fast“ serverto two or three ”slower“ servers even if no decrease insystem utilization results. This occurs because a one-to-one correspondence between line length and themeans of the distribution of threshold values results ina scenario in which waits grow faster than customer

Table 5 Percentage of Satisfied Customers of Hidden and Revealed M/M/c Models for Various Levels of System Utilization for Various Values of c

Hidden M/M/c with X � Exp(�)

Number of Servers c

Utilization �

0.20 0.50 0.80 0.90 0.98

1 0.8889 0.6667 0.3333 0.1818 0.03923 0.9863 0.8421 0.4607 0.2572 0.05646 0.9997 0.9752 0.7646 0.5374 0.1561

Revealed M/M/c with Yn � Exp(�/n)

Number of Servers c

Utilization �

0.20 0.50 0.80 0.90 0.98

1 0.8976 0.7320 0.5090 0.3474 0.09433 0.9932 0.9223 0.7095 0.5161 0.15316 0.9997 0.9787 0.8064 0.5821 0.1661

* For all cases c � 100 and � � c.** The reported numbers are the percentage of satisfied customers F(0).


patience when they view and wait in longer lines. Theresult is that even though pooling results in less wait-ing, it can also result in a lower PSC because custom-ers are waiting in longer lines.

In summary, our approach emphasizes the fact thatthe objective of increasing customer satisfaction canprofitably be approached by altering the service sys-tem to ”provide more service“ or by altering the cus-tomers’ attitude toward the waiting experience. Wehave developed and presented a number of relativelysimple analytic models that managers can use to eval-uate the impact of policies which change the servicedelivery system or customer thresholds. These ideasmay prove quite useful in the design and managementof service delivery systems.

AppendixProof of Lemma 1.

P�I � J � 0� � �0

�

G1 � y��e��yd y � ��

�� .

F(0) for Hidden Queues:For a stationary M/G/1 system, the inversion formulaof the Pollaczek-Khinchine (P-K) waiting time trans-form is known. (See Kleinrock 1975, p. 201, equation5.111.)

h�t� � �n�0

�

�1 � ��nr�n��t� (A1)

where r(t) is the stationary residual service time den-sity given by

r�t� �ddt R�t� and R�t� �

1m1

�1 � B�t�� (A2)

and r(n) represents the n-fold convolution of r withitself.

We can evaluate h(t) numerically by applying theLaguerre transform to equations (A1) and (A2). For adetailed reference on the use of Laguerre transforma-tion, see Keilson and Nunn (1979). When the distribu-tion function of X is Gamma, hyperexponential, foldednormal, their convolution, their mixture, or combina-tion, it is convenient to evaluate the bilateral Laguerretransform of W – X numerically to describe the distri-bution of the random variable Z. For a detailed refer-ence describing this approach, see Keilson, Nunn, andSumita (1981).

F(0) for Revealed QueuesRecall that Yn is the threshold value given that npeople are in the line when the new customer arrives.We must consider the following distribution function:Fn(x) � P{W � Yn � 0�L � n}. In particular, we are

interested in calculating Fn(0), which represents theprobability that a customer arriving in the systemwhen there are n customers ahead of him/her will besatisfied, i.e., W � Yn. Since the randomness of Ynarises from the heterogeneity of the customer popula-tion, we assume that Yn and W are conditionally in-dependent given L. Therefore, we fix Y0 � 0.

Suppose that a customer observes n customersahead of him/her at the time of arrival. Because the”Poisson arrival sees time averages“ (Wolff 1982), theresidual service time of the customer being served, ifany, at a customer arrival epoch has the distributionfunction R(t) given in (2). Thus, the distribution func-tion of the waiting time given that an arriving cus-tomer sees n customers in the system is

Hn � x� � P�W � x�L � n� � � B(n�1)�R( x) n � 1U( x) n � 0 ,

(A3)

where B(n) is n-fold convolution of B with itself,* represents a convolution operation, and U(x) is thestep function previously defined.

When the distribution function of Yn is Gamma,hyperexponential, folded normal, their convolution,their mixture, or combination of those, the distributionfunction Hn(x) can be numerically evaluated using thebilateral Laguerre transform (see Keilson, Nunn, andSumita 1981).

To complete the calculation of (5), we also need thestationary queue length distribution (qn)n�0

� of ourM/G/1 queue. Its calculation is based on the classicalPollaczek-Khinchine queue length formula (see Klein-rock 1975):

Q� z� � �n�0

�

qn zn � B�� z��1 � ��1 � z�

B�� z� � z,

(A4)

where B(s) is the L-S transform of B(x). Let us assumethat the service time distribution function has a ratio-nal L-S transform. It is known that this class of distri-bution functions is ”dense“ (Kingman 1966) in thesense that every distribution function can be well ap-proximated by a distribution function with rationalL-S transform. Therefore, our assumption does notimpose any serious restrictions in practice. In this case,B(� � �z) � (N(z)/D(z)). We note that N(z) and D(z)are both polynomials. From (A4) we obtain,

Q� z� ��1 � �� N� z��1 � z�

N� z� � zD� z�. (A5)

To invert (A3), we apply a straightforward powerseries expansion. Let

q0 � Q�0� and Q0�z� � Q�z�. (A6)


Given Qn(z) and qn, let

Qn 1( z) �Qn( z) � qn

zqn 1 � Qn 1(0)

� . (A7)

Then, the recursive formula (A7), with the initial con-dition (A6) gives the series expression of (A4). Fromthe series expression and (5), we can evaluate F(x).

F(0) with Balking and Reneging for M/M/1 QueuesThe analysis of both hidden and revealed queues forsuch settings is very similar, thus their treatment ispresented together here. Balking results in an effectivearrival rates that take into account customers whoabandon the queue before joining for any significantlength of time. We label this rate �n � �bn where 0 �bn � b0 � 1. When the length of the queue is hidden,we can assume that bi is a constant b for all i � 0 inhidden queues. For M/M/1 systems, we know (seeGross and Harris 1998, p. 94) that,

qn � q0 �i�1

n�i�1

� q0��

�n �

i�1

n

bi�1 . (A8)

For cases with reneging customers, we follow the ap-proach in Gross and Harris (1998), p. 95, to devise qnvalues where reneging rates (r(n)) are functions of n.Combining the effects of balking and reneging on qnwe have,

qn � q0�n �

i�1

n bi�1

� r�n��n � 1�, and q0

� �1 � �n�1

� ��n �i�1

n bi�1

� r(n)��1

. (A9)

We calculate the percentage of customers that balk as

� � �n�1

�

qn �1 � bn �. (A10)

When customers renege at a constant rate, r(n) � ,this is equivalent to having customers leave the sys-tem after exponentially distributed amounts of timewith rate . When a customer finds n � c k custom-ers in the system, the delay can be represented as thesum of k 1 independent exponential random vari-ables. Note that these variables are not identicallydistributed unless � . Reneging occurs at some rate k for each position in the queue and other customersleave the system after being served at some rate . Thetotal reneging rate when there are k � n – c customers

in the queue is k � ¥j�1k j. Thus, the waiting time

distribution is the sum of k 1 independent exponen-tial random variables with different means. The L-Stransform of this distribution can be written as

Wk 1 �s� � �j�0

k 1 � � j

� j � s� . (A11)

For hidden systems when customers renege at a con-stant rate, j is independent of j, so we may drop thesubscript. For the case of a single server, application ofLemma 1 implies,

Fn �0� � Wn ��

� � ��n

. (A12)

Wn( ) given by (A11) states the percentage of cus-tomers who enter a system containing n customersand do not balk, nor renege. If we label the percentageof arriving customers that do not balk or renege andare satisfied as F�(0), we can now state that,

F��0� � �n�0

�

qn bn� � j

� 2 j� n

Fn �0�. (A13)

We assume that customers who balk or renege arenot satisfied. We label the percentage of reneging cus-tomers �. We can now state � and PSC � F(0) as,

� � �n�1

�

bn qn�1 � Wn � ��, and (A14)

F�0� � F��0��1 � � � ��. (A15)

For additional details, see Rao (1968).

F(0) for Hidden M/M/c QueueWe know that F�0� � ¥n�0

c�1 qn ¥n�c� qn�c/�c ��n�c 1.

Expansion and rearranging of terms yields, F(0)� ¥n�0

c�1q0(�n/n!n) ¥n�c

� q0(�ncc/nc!cn)(c/(c �))n�c 1

� q0 ¥n�0c�1 ((�/)n) (1/n!) q0 ¥n�c

� (�ncc/c!)(c/(c � ))n�c 1, where, � � (�/c). Therefore, F(0) � q0¥n�0

c�1 ((c�)n/n!) q0 ((�c)c/c!) (c/(c � )) ¥n�0� (�

(c/(c � ))n. Simplification leads to F(0) � q0 ¥n�0c�1

((c�)n/n!) q0 ((�c)c/c!)(c/(c �))(c �)/(c ��).Rearranging terms leads to the stated form.

Proof of Proposition 4. If this is true for � � 1, it isalso true for all 0 � � � 1. Setting � � 1, and �n � �/n,then according to (10), we have

Fn �0� � � n

n � ��n

.

Then


�nFn �0� � n��nn � �n�n � � �� 3d�nFn �0�

dn

� �nn � �n�n � � �n�

n�

(n � �)� 3 dFn �0�

dnFn �0�

� �nn � �n�n � � � ��

�n � � �.

Let En � �nn � �n(n � ) (�/(n � )). Thenlimn30 En � �� and limn30 En � 0. Since

dEn

dn �mn

�

�n � � ��

�

�n � � �2

��2

n�n � � �2 � 0,

En is a monotone increasing function and En � 0 for alln � 0. Thus,

dFn �0�

dn � 0 for all n � 0.

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