G lossary

puted, total expected costs were studied. As shown graphically in Figure 14.1, total cost isthe sum of the cost of providing service plus the cost of waiting time.

Key operating characteristics for a system were shown to be (l) utilization rate, Hereisalistingoftheke(2) percent idle time, (3) average time spent waiting in the system and in the queue, (4) systemcharacteristicsaverage number of customers in the system and in the queue, and (5) probabilities of vari-ous numbers of customers in the system.

It was emphasized that a variety of queuing models exist that do not meet all of theassumptions of the traditional models. In these cases w use more cornplex mathematicalmodels or turn to a technique called computer simulation. The application of simulation toproblems. of queuing systems, inventory control, machine breakdown, and other quantita-tive analysis situations is the topic discussed in Chapter 15.

Waiting Line. One or more customers or objects waiting to be served:Queuing Theory. The mathematicalstudy of waiting lines or queues.Service Cost. The cost ofproviding a particular level ofservice.Waiting Cost. The cost to the firm of having customers or objects waiting in line to be serviced.Calling Population. The population of items from which arrivals at the queuing system come.Unlimited or Infinite Population. A calling population that is very large relative to the number of

customers currently in the system.Limited or Finite Population. A case in which the number of customers in the system is a signifi-

cant proportion of the calling population.Poisson Distribution. A probability distribution that is often used to describe random arrivals in a

queue.

Balking. The case in which aniving customers refuse to join the waiting line.Reneging. The case in which customers enter a queue but then leave before being serviced.Limited Queue Length. A waiting line that cannot increase beyond a specific size.Unlimited Queue Length. A queue that can increase to an infinite size.Queue Discipline. The rule by which customers in a line receive service.FIFO. A queue discipline (meaning first-in, first-out) in which the customers are served in the strict

order of arrival.

Single-Channel Queuing System. A system with one service facility fed by one queue.Multiple-Channel Queuing System. A system that has more than one service facility, all fed by

the same single queue.Single-Phase System. A queuing system in which service is received at only one station.Multiphase System. A system in which service is received from more than one station, one after

the other.

Negative Exponential Probability Distribution. A probability distribution that is often used todescribe random service times in a service system.

M/[d/l. Another name for the single-channel model with Poisson arrivals and exponential servicetimes.

Operating Characteristics. Descriptive characteristics of a queuing system, including the averagenumber of customers in a line and in the system, the average waiting times in a line and in thesystem, and percent idle time.

640 Chapter 14 Wntrtrue Ltrurs nruo Queutruc THronv Moost-s

Utitization Factor (p). The proportion of the time that ser-vice facilities are in use.M/M/m. A technical name for the multichannel queuing model (with nr servers) and Poisson ar-

rivals and exponential service times.MIDll, A technical name for the constant service time model.Simulation. A technique for qqpresenting queuing rnodels that are complex and difficult to model

analytically.

tr : mean number of arrivals per time periodp : mean number of people or items served per time period

(14-r) P(D =+Poisson probability distribution used in describidg arrivals.

Equations 14-2 through l4-8 describe operating characteristics in the single-channel model thathas Poisson anival and exponential sertice rates.(14-2') f, : average number of units (customers) in the system

tL I(14-3) W = aveftgetime a unit spends in the system (waiting time * service time)

I=-1t-t

(14-4) Lo = avengenumber of units in the queue : R#"(14-5) Wo : average time a unit spends waiting in the queue : mb(14-6) P = utilization factor for the system = !'lL{14-7) Po = probabiliry of 0 units in the system (that is, the service unit is idle)

=t-rlL

(L4-8) P,>k= probability of more than & units in the system : (t)--tEquations l4-9 through 14-14 describe operating characteristics in multipk-chatnel models thnthave Poisson arrival and. exponential service rates, where M : the number ofopen qhanncls'

(14-9) Po :

The probability thai there are no people ot umts in tlrc system'

- ),,p(Ilp.)M o-'tr(14-10) L : W _ r)t-di=E ro + i

The average number of people or units in the system.

p(Ild-o -l:L(r4-rr) w = W=ffii _-I:J_ro * i: i

The average time a unit spends in the waiting lin ort"ir,g ,"*l*Od;;y, in the system).

for Mp,> A

Key Equations

(t4-12) L":L-L'11

The average number of people o, unir, in line waiting for service.lLn(r4-13) w": w

' p.- ).The average time a person or unit spends in the queue waiting for service.

(t4-t4) r:hUtilization rate.

Equations 14-15 through 14-18 describe operating characteristics in single-channel models thathave Poisson arrivals and constant service rates.

(14-rs) Lq: ztrd_ ^)

. The average length of the queue.

I(14-16l W : ---L-_, ztt"(tt

- )t)

The average waiting time in the queue.

(14-17) L= L^ + LlL

The average number of customers in the system.

(14-18) W: W^ + L'lL

The average waiting time in the system.

Equations 14-19 through 14-24 describe operating characteristics in single-channel models thathave Poisson arrivals and exponential sewice rates and afinite cailing poputatian.

(14-19) Po:

641

:"5(t)"The probability that the sysrem is empty.

(t4-zo) Lo : N- fL+*) (l - P")' \^ /Average length of the queue.

(14-21) L: Lo + (1 -

Po)Average number of units in the system. .

L{t4-22) wq: av +)i

Average time in the queue.1(14-23) w: wq + i

Average time in the system.

{t4-zr) P": (N$(t)" "

ror n : 0,r,. ., NProhahilitv of n unifs in flre svsfem

01-queuing glossary02-queuing formula03- queuing formula