Service OperationsService Operationsandand

Waiting LinesWaiting Lines

Dr. Everette S. Gardner, Jr.

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Case study: Single-server modelReference

Vogel, M. A., “Queuing Theory Applied to Machine Manning,”Interfaces, Aug. 79.

CompanyBecton - Dickinson, mfg. of hypodermic needles and syringes

Bottom lineCash savings = $575K / yr.Also increased production by 80%.

ProblemHigh-speed machines jammed frequently. Attendants cleared jams. How many machines should each attendant monitor?

ModelBasic single-server:

Server—AttendantCustomer—Jammed machine

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Case study (cont.)

Solution procedureEach machine jammed at rate of λ = 60/hr.

With M machines, arrival rate to each attendant is λ = 60M

Service rate is μ = 450/hr.

Utilization ratio = 60M/450

Experimenting with different values of M produced an arrival rate that minimized costs (wages + lost production)

M = 5 was optimal, compared to M = 1 before queuing study

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Case study: Multiple-server modelReference

Deutch, H. and Mabert, V. A., “Queuing Theory Applied to Teller Staffing,” Interfaces, Oct., 1980.

CompanyBankers Trust Co. of New York

Bottom lineAnnual cash savings of $1,000,000 in reduced wages. Cost to develop model of $110,000.

ProblemDetermine number of tellers to be on duty per hour of day to meet goals for waiting time. Staffing decisions needed at 100 branch banks.

ModelStraightforward application of multi-channel model in text.

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Case study (cont.)Analysis

Development of arrival and service distributions by hour and day of week at each bank.

Arrival and service shown to be Poisson / Exponential.

Experimentation with number of servers in model showed that full-time tellers were idle much of the day.

ResultElimination of 100 full-time tellers. Increased use of part-

time tellers.

Today, the multi-channel model is a standard tool for staffing decisions in banking.

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Queuing model structures

Single-server model

Pop. Arrival Queue Service time can be rate capacity can usually

exp., finite or must be be finite but can be infinite Poisson or infinite anything

Sourcepop.

Servicefacility

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Queuing model structures (cont.)Multiple-server model

Pop. Arrival Queue must be rate capacity infinite must be must be

Poisson infinite Service time for each

Note: There is only one queue server must

regardless of nbr. of servers have same mean and be exp.

Sourcepop.

Servicefacility

#1

Servicefacility

#2

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Applying the single-server model1. Analyze service times.

- plot actual vs. exponential distribution- if exponential good fit, use it- otherwise compute σ of times

2. Analyze arrival rates.- plot actual vs. Poisson Distribution- if Poisson good fit, use it- if not, stop—only alternative is simulation

3. Determine queue capacity.- infinite or finite?- if uncertain, compare results from alternative models

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Applying the single-server model (cont.)

4. Determine size of source population.- infinite or finite?- if uncertain, compare results from alternative models

5. Choose model from SINGLEQ worksheet.

SINGLEQ.xls

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Applying the multiple-server model

1. Analyze service times. - Must be exponential

2. Analyze arrival rates. - Must be Poisson

3. Queue capacity must be infinite.

4. Source population must be infinite.

5. Apply MULTIQ worksheet.

MULTIQ.xls

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Single-server equations

Arrival rate = λ

Service rate = μ

Mean number in queue = λ2/(μ(μ-λ))

Mean number in system = λ /(μ-λ)

Mean time in queue = λ /(μ(μ-λ))

Mean time in system = 1/(μ-λ)

Utilization ratio = λ /μ(Prob. server is busy)

SINGLEQ.xls

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Utilization ratio vs. queue length

λ μ λ/μ Queue length 5 20 .25 0.08 people10 20 .50 0.5015 20 .75 2.2519 20 .95 18.0519.5 20 .975 38.0319.6 20 .98 48.0219.7 20 .985 64.6819.8 20 .99 98.0119.9 20 .995 198.0119.95 20 .997 398.0019.99 20 .999 1,998.00

20 20 1.000 SINGLEQ.xls

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Single-server queuing identities

A. Number units in system = arrival rate * mean time in system

B. Number units in queue = arrival rate * mean time in queue

C. Mean time in system = mean time in queue + mean service time

Note: Mean service time = 1/ mean service rate

If we can determine only one of the following, all other values can befound by substitution:

Number units in system or queueMean time in system or queue

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State diagram: single-server model

A A A

# in system

S S S

● # in system also called state.

● To get from one state to another, an arrival (a) must occur or a service completion (s) must occur.

● In long-run, for each state:Rate in = Rate outMean # A = Mean # S

3210

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Balance equations for each state

State Rate in = Rate out 0 SP1 AP0

Probability in Probability in state 1 state 0

The only way The only way into state 0 out of state 0

is service is to have completion from 1 an arrival

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Balance equations for each state (cont.)State Rate in = Rate out 1 AP0 + SP2 = AP1 + SP1

Can arrive Two ways state 1 by out of state 1, arrival from 0 arrival or or service service completion completion from 2

2 AP1 + SP3 = AP2 + SP2

3 AP2 + SP4 = AP3 + SP3

etc.

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Solution of balance equations

Expected number in system = ΣnPn

Solve equations simultaneously to get each probability.

Given number in system, all other values are found by substitution in queuing identities.