Think about an activity where you frequently wait in line (restaurant, sporting venues, retail outlets, etc.) Describe the

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The Waiting Line Problem

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LO10-1

Understand what a waiting line problem is.

A central problem in many service settings is the management of waiting time. The manager must weigh the added cost of providing more rapid

service (more traffic lanes, additional landing strips, more checkout stands) against the inherent cost of waiting.

Frequently, the cost trade-off decision is straightforward. For example, if we find

that the total time our employees spend in line waiting to use a copying machine

would otherwise be spent in productive activities, we could compare the cost of installing one additional machine to the value of employee time saved. The decision

could then be reduced to dollar terms and the choice easily made.

On the other hand, suppose that our waiting line problem centers on demand for beds in a hospital. We can compute the cost of additional beds by summing the

costs for building construction, additional equipment required, and increased

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maintenance. But what is on the other side of the scale? Here, we are confronted

with the problem of trying to place a dollar figure on a patient’s need for a hospital bed that is unavailable. While we can estimate lost hospital income, what about the

human cost arising from this lack of adequate hospital care?

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The Practical View of Waiting Lines Before we proceed with a technical presentation of waiting line theory, it is useful to

look at the intuitive side of the issue to see what it means. Exhibit 10.1 shows

arrivals at a service facility (such as a bank) and service requirements at that facility (such as tellers and loan officers). One important variable is the number of arrivals

over the hours that the service system is open. From the service delivery viewpoint,

customers demand varying amounts of service, often exceeding normal capacity. We can control arrivals in a variety of ways. For example, we can have a short line

(such as a drive-in at a fast-food restaurant with only several spaces), we can establish specific hours for specific customers, or we can run specials. For the

server, we can affect service time by using faster or slower servers, faster or slower

machines, different tooling, different material, different layout, faster setup time, and so on.

EXHIBIT 10.1 Arrival and Service Profiles

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The essential point is waiting lines are not a fixed condition of a productive system

but are to a very large extent within the control of the system management and design. The following are useful suggestions for managing queues

based on research in the banking industry:

Segment the customers. If a group of customers need something that can be done very quickly, give them a special line so they do not have to wait for the slower

customers. This is commonly done at grocery stores where checkout lines are

designated for “12 items or less.”

Train your servers to be friendly. Greeting the customer by name or providing another form of special attention can go a long way toward overcoming the

negative feeling of a long wait. Psychologists suggest that servers be told when to invoke specific friendly actions such as smiling when greeting customers, taking

orders, and giving change (for example, in a convenience store). Tests using such

specific behavioral actions have shown significant increases in the perceived friendliness of the servers in the eyes of the customer.

Inform your customers of what to expect. This is especially important when the

waiting time will be longer than normal. Tell them why the waiting time is longer

than usual and what you are doing to alleviate the wait.

Try to divert the customer’s attention when waiting. Providing music, a video, or some other form of entertainment may help distract the customers from the fact

that they are waiting.

Encourage customers to come during slack periods. Inform customers of times

when they usually would not have to wait; also tell them when the peak periods are—this may help smooth the load.

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The Queuing System The queuing system consists essentially of three major components: (1) the source population and the way customers arrive at the system, (2) the servicing system,

and (3) the condition of the customers exiting the system (back to source

population or not?), as seen in Exhibit 10.2. The following sections discuss each

of these areas.

Customer Arrivals Arrivals at a service system may be drawn from a finite or an infinite population. The distinction is important because the analyses are based on

different premises and require different equations for their solution.

EXHIBIT 10.2 Components of a Queuing System

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Page261Finite Population A finite population refers to the limited-size customer pool that will use the service and, at times, form a line. The reason this

finite classification is important is that when a customer leaves their position as a

member of the population (due to a machine breaking down and requiring service, for example), the size of the user group is reduced by one, which reduces the

probability of the next occurrence. Conversely, when a customer is serviced and returns to the user group, the population increases and the probability of a user

requiring service also increases. This finite class of problems requires a separate set

of formulas from that of the infinite population case.

As an example, consider a group of six machines maintained by one repairperson. In this case, the machines are the customers of the repairperson. When one

machine breaks down, the source population is reduced to five, and the chance of one of the remaining five breaking down and needing repair is certainly less than

when six machines were operating. If two machines are down with only four operating, the probability of another breakdown is again changed. Conversely, when

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a machine is repaired and returned to service, the machine population increases,

thus raising the probability of the next breakdown.

Infinite Population An infinite population is large enough in relation to the service system so that the population size caused by subtractions or additions to the

population (a customer needing service or a serviced customer returning to the population) does not significantly affect the system probabilities. If, in the preceding

finite explanation, there were 100 customer machines instead of 6, then if 1 or 2

machines broke down, the probabilities for the next breakdowns would not be very different and the assumption could be made without a great deal of error that the

population (for all practical purposes) was infinite. Nor would the formulas for “infinite” queuing problems cause much error if applied to a physician with 1,000

patients or a department store with 10,000 customers.

Distribution of Arrivals When describing a waiting system, we need to define the manner in which customers or the waiting units are arranged for service.

Waiting line formulas generally require an arrival rate, or the number of units per

period (such as an average of one every six minutes). A constant arrival distribution

is periodic, with exactly the same time between successive arrivals. In productive systems, the only arrivals that truly approach a constant interval period are those

subject to machine control. Much more common are variable (random) arrival distributions.

In observing arrivals at a service facility, we can look at them from two viewpoints:

First, we can analyze the time between successive arrivals to see if the times follow

some statistical distribution. Usually, we assume that the time between arrivals is exponentially distributed. Second, we can set some time length (T) and try to

determine how many arrivals might enter the system within T. We typically assume that the number of arrivals per time unit is Poisson distributed.

Exponential Distribution In the first case, when arrivals at a service facility occur in a purely random fashion, a plot of the interarrival times

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yields an exponential distribution such as that shown in Exhibit 10.3. The

probability function is

EXHIBIT 10.3 Exponential Distribution

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[10.1]

where λ is the mean number of arrivals per time period.

The cumulative area beneath the curve in Exhibit 10.3 is the summation of

Equation 10.1 over its positive range, which is This integral allows us to

compute the probabilities of arrivals within a specified time. For example, for the case of one arrival per minute to a waiting line the following table can be

derived either by solving or by using Appendix B. Column 2 shows the probability that it will be more than t minutes until the next arrival, while column 3

shows the probability of the next arrival within t minutes (computed as 1 minus

column 2).

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Poisson Distribution In the second case, where one is interested in the number of

arrivals during some time period T, the distribution appears as in Exhibit 10.4 and is obtained by finding the probability of exactly n arrivals during T. If the arrival

process is random, the distribution is the Poisson, and the formula is

(1) (2)

t (Minutes)

Probability that the Next Arrival Will Occur in

t Minutes or More (from Appendix B or Solving e−t)

Probability that t

t Minutes o

0 100%

0.5 61

1.0 37

1.5 22

2.0 14

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EXHIBIT 10.4 Poisson Distribution for λT = 3

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[10.2]

Equation 10.2 shows the probability of exactly n arrivals in time T. (Note that n! is

defined as n(n-1)(n-2)…(2)(1).) For example, if the mean arrival rate of units into a

system is three per minute (λ = 3) and we want to find the probability that exactly five units will arrive within a one-minute period (n = 5, T = 1), we have

That is, there is a 10.1 percent chance that there will be five arrivals in any one-

minute interval.

Although often shown as a smoothed curve, as in Exhibit 10.4, the Poisson is a discrete distribution. (The curve becomes smoother as n becomes large.) The

distribution is discrete because, in our example, n refers to the number of arrivals in a system, and this must be an integer. (For example, there cannot be 1.5 arrivals.)

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Also note that the exponential and Poisson distributions can be derived from one

another. The mean and variance of the Poisson are equal and denoted by λ. The

mean of the exponential is 1/λ and its variance is 1/λ2. (Remember that the time

between arrivals is exponentially distributed and the number of arrivals per unit of time is Poisson distributed.)

Other arrival characteristics include arrival patterns, size of arrival units, and

degree of patience. (See Exhibit 10.5.)

EXHIBIT 10.5 Customer Arrivals in Queues

Arrival patterns. The arrivals at a system are far more controllable than is

generally recognized. Barbers may decrease their Saturday arrival rate (and supposedly shift it to other days of the week) by charging an extra $1 for adult

haircuts or charging adult prices for children’s haircuts. Department stores run

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Waiting Lines and Servers The queuing system consists primarily of the waiting line(s) and the available number of servers. Here, we discuss issues pertaining to waiting line characteristics and management, line structure, and service rate.

sales during the off-season or hold one-day-only sales in part for purposes of

control. Airlines offer excursion and off-season rates for similar reasons. The simplest of all arrival-control devices is the posting of business hours.

Some service demands are clearly uncontrollable, such as emergency medical demands on a city’s hospital facilities. But even in these situations, arrivals at

emergency rooms in specific hospitals are controllable to some extent by, say,

keeping ambulance drivers in the service region informed of the status of their respective host hospitals.

Size of arrival units. A single arrival may be thought of as one unit. (A unit is the

smallest number handled.) A single arrival on the floor of the New York Stock

Exchange (NYSE) is 100 shares of stock; a single arrival at an egg-processing plant might be a dozen eggs or a flat of 2½ dozen; a single arrival at a restaurant

is a single person. A batch arrival is some multiple of the unit, such as a block of 1,000 shares on

the NYSE, a case of eggs at the processing plant, or a party of five at a

restaurant.

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Degree of patience. A patient arrival is one who waits as long as necessary until the service facility is ready to serve him or her. (Even if arrivals grumble and

behave impatiently, the fact that they wait is sufficient to label them as patient arrivals for purposes of waiting line theory.)

There are two classes of impatient arrivals. Members of the first class

arrive, survey both the service facility and the length of the line, and then decide to leave. Those in the second class arrive, view the situation, join the waiting line,

and then, after some period of time, depart. The behavior of the first type is termed balking, while the second is termed reneging. To avoid balking and

reneging, companies that provide high service levels typically try to target server

utilization levels (the percentage of time busy) at no more than 70 to 80 percent.

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Factors to consider with waiting lines include the line length, number of lines, and

queue discipline.

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Length. In a practical sense, an infinite line is simply one that is very long in terms of the capacity of the service system. Examples of infinite potential length are a line of vehicles backed up for miles at a bridge crossing and customers who must form a line around the block as they wait to purchase tickets at a theater.

Gas stations, loading docks, and parking lots have limited line capacity caused by legal restrictions or physical space characteristics. This complicates the waiting line problem not only in service system utilization and waiting line computations but also in the shape of the actual arrival distribution. The arrival denied entry into the line because of lack of space may rejoin the population for a later try or may seek service elsewhere. Either action makes an obvious difference in the finite population case.

Number of lines. A single line or single file is, of course, one line only. The term multiple lines refers to the single lines that form in front of two or more servers or to single lines that converge at some central redistribution point. The disadvantage of multiple lines in a busy facility is that arrivals often shift lines if several previous services have been of short duration or if those customers currently in other lines appear to require a short service time.

Queue discipline. A queue discipline is a priority rule or set of rules for determining the order of service to customers in a waiting line. The rules selected can have a dramatic effect on the system’s overall performance. The number of customers in line, the average waiting time, the range of variability in waiting time, and the efficiency of the service facility are just a few of the factors affected by the choice of priority rules.

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Probably the most common priority rule is first come, first served (FCFS). This rule states that customers in line are served on the basis of their chronological arrival; no other characteristics have any bearing on the selection process. This is popularly accepted as the fairest rule, although in practice it discriminates against the arrival requiring a short service time.

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Reservations first, emergencies first, highest-profit customer first, largest orders first, best customers first, longest waiting time in line, and soonest promised date are other examples of priority rules. There are two major practical problems in using any rule: One is ensuring that customers know and follow the rule. The other is ensuring that a system exists to enable employees to manage the line (such as take-a-number systems).

Service Time Distribution Another important feature of the waiting structure is the time the customer or unit spends with the server once the service has started. Waiting line formulas generally specify service rate as the capacity of the server in

number of units per time period (such as 12 completions per hour) and not as

service time, which might average five minutes each. A constant service time rule states that each service takes exactly the same time. As in constant arrivals, this

characteristic is generally limited to machine-controlled operations.

When service times are random, they can be approximated by the exponential distribution. When using the exponential distribution as an approximation of the

service times, we will refer to μ as the average number of units or customers that can be served per time period.

Line Structures As Exhibit 10.6 shows, the flow of items to be serviced may go through a single line, multiple lines, or some mixture of the two. The choice of

format depends partly on the volume of customers served and partly on the restrictions imposed by sequential requirements governing the order in which

service must be performed.

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EXHIBIT 10.6 Line Structures

1. Single channel, single phase. This is the simplest type of waiting line structure,

and straightforward formulas are available to solve the problem for standard distribution patterns of arrival and service. When the distributions are

nonstandard, the problem is easily solved by computer simulation. A typical

example of a single-channel, single-phase situation is the one-person barbershop.

2. Single channel, multiphase. A car wash is an illustration because a series of services (vacuuming, wetting, washing, rinsing, drying, window cleaning, and

parking) is performed in a fairly uniform sequence. A critical factor in the single-

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channel case with service in series is the amount of buildup of items allowed in

front of each service, which in turn constitutes separate waiting lines.

Page2663. Multichannel, single phase. Tellers’ windows in a bank and checkout counters in high-volume department stores exemplify this type of

structure. The difficulty with this format is that the uneven service time given

each customer results in unequal speed or flow among the lines. This results in some customers being served before others who arrived earlier, as well as in

some degree of line shifting. Varying this structure to ensure the servicing of arrivals in chronological order would require forming a single line, from which,

as a server becomes available, the next customer in the queue is assigned.

The major problem of this structure is that it requires rigid control of the line to maintain order and to direct customers to available servers. In some instances,

assigning numbers to customers in the order of their arrival helps alleviate this problem.

4. Multichannel, multiphase. This case is similar to the preceding one except that two or more services are performed in sequence. The admission of patients in a

hospital uses this pattern because a specific sequence of steps is usually followed: initial contact at the admissions desk, filling out forms, making identification

tags, obtaining a room assignment, escorting the patient to the room, and so forth. Because several servers are usually available for this procedure, more than

one patient at a time may be processed.

5. Mixed. Under this general heading, we consider two subcategories: (1) multiple-

to-single channel structures and (2) alternative path structures. Under (1), we find either lines that merge into one for single-phase service, as at a bridge

crossing where two lanes merge into one, or lines that merge into one for

multiphase service, such as subassembly lines feeding into a main line. Under (2), we encounter two structures that differ in directional flow requirements. The

first is similar to the multichannel–multiphase case, except that (a) there may be switching from one channel to the next after the first service has been rendered

and (b) the number of channels and phases may vary—again—after performance

of the first service.

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Page267Exiting the Queuing System Once a customer is served, two exit fates are possible: (1) The customer may return to the source population and immediately become a competing candidate for service again or (2) there may be a

low probability of reservice. The first case can be illustrated by a customer machine that has been routinely repaired and returned to duty but may break down again; the

second can be illustrated by a customer machine that has been overhauled or

modified and has a low probability of reservice over the near future. In a lighter vein, we might refer to the first as the “recurring-common-cold case” and to the

second as the “appendectomy-only-once case.”

It should be apparent that when the population source is finite, any change in the

service performed on customers who return to the population modifies the arrival rate at the service facility. This, of course, alters the characteristics of the waiting

line under study and necessitates reanalysis of the problem.