# Determine the maximum line length for a probability of 91 percent

The following information pertains to telephone calls to a motel switchboard on a typical

Tuesday. Use Table 1.

Period

Incoming

Rate (calls

per minute)

Service

Rate (calls

per minute

per

operator)

Number of

Operators

Morning 1.6 1.6 2

Afternoon 2.7 .9 4

Evening 1.5 .6 3

a.

Determine the average time callers wait to have their calls answered for each period and the

probability that a caller will have to wait for each period. (Round intermediate calculations

and final answers to 3 decimal places.)

Period Pw

Morning

Afternoon

Evening

b.

Determine the maximum line length for a probability of 91 percent. (Round your

intermediate calculations to 3 decimal places and round up your final answers to the

next whole number.)

Period Lmax

Morning

Afternoon

Evening

Trucks are required to pass through a weighing station so that they can be checked for weight

violations. Trucks arrive at the station at the rate of 24 an hour between 7:00 p.m. and 9:00 p.m.

Currently two inspectors are on duty during those hours, each of whom can inspect 15 trucks an

hour.

Use Table 1.

a.

How many trucks would you expect to see at the weighing station, including those being

inspected? (Round your answer to 3 decimal places.)

Ls

trucks

b.

If a truck was just arriving at the station, about how many minutes could the driver expect to

be at the station? (Round your answer to 2 decimal places.)

Ws

min.

c.

What is the probability that both inspectors would be busy at the same time?(Round your

answer to 4 decimal places.)

Pw

d.

How many minutes, on average, would a truck that is not immediately inspected have to

wait? (Round your answer to the nearest whole number.)

Wa

min.

f.

What is the maximum line length for a probability of .90? (Round up your answer to the

next whole number.)

L max

banking hours. During the early evening hours in the summer months, customers arrive at a

certain location at the rate of one every other minute. This can be modeled using a Poisson

distribution. Each customer spends an average of 90 seconds completing his or her transactions.

Transaction time is exponentially distributed.

a.

Determine the average time customers spend at the machine, including waiting in line and

completing transactions. (Do not round intermediate calculations. Round your answer to

the nearest whole number.)

Average time minutes

b.

Determine the probability that a customer will not have to wait upon arriving at the automatic

teller machine. (Round your answer to 2 decimal places.)

Probability

c.

Determine the average number waiting to use the machine. (Round your answer to 2

decimal places.)

Average number customers

constant. Customers arrive at a mean rate of 75 per hour, and this rate is Poisson-distributed.

a.

Determine the average number of customers waiting in line. (Round your answer to 2

decimal places.)

Average number of customer

b.

Determine the average time customers spend in the system. (Do not round intermediate

calculations. Round your answer to 2 decimal places.)

Average time minutes

c.

Determine the average number in the system. (Do not round intermediate calculations.

Round your answer to 2 decimal places.)

Average number

Repair calls are handled by one repairman at a photocopy shop. Repair time, including travel

time, is exponentially distributed, with a mean of 2.5 hours per call. Requests for copier repairs

come in at a mean rate of 2.1 per eight-hour day (assume Poisson).

_2013_QC_28691

a.

Determine the average number of customers awaiting repairs. (Round your answer to 2

decimal places.)

Number of customers

b.

Determine system utilization. (Round your answer to the nearest whole percent. Omit the

“%” sign in your response.)

System utilization %

c.

Determine the amount of time during an eight-hour day that the repairman is not out on a

call. (Round your answer to 2 decimal places.)

Amount of time hours

d.

Determine the probability of two or more customers in the system. (Do not round

intermediate calculations. Round your answer to 4 decimal places.)

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