Queuing



Queuing

What is queuing theory?

• The mathematical approach to the analysis of lines

• Useful in planning and analysis of service capacity

• Goal of queuing -- minimize total cost - costs associated with customers waiting in line for service and those associated with capacity

System Characteristics

1) Population source

o Infinite source

o Finite source

Number of servers (channels)

Single

Multiple

Arrival and service patterns

Probability distribution (exponential, Poisson, etc)

Queue discipline (order of service)

First-come-first-served

Queuing Models:

Infinite Sources

• Assumptions:

o Poisson arrival rate

o System operates under steady state (average arrival and service rates are stable)

Important note: The arrival (() and service rates (() must be in the same units

Four Basic Models

1) Single channel, exponential service time

2) Single channel, constant service time

3) Multiple channel, exponential service time

4) Multiple priority service, exponential service time

Finite Source

1) Appropriate for cases in which the calling population is limited to a relatively small number of potential calls

2) Example -- one person may be responsible for handling breakdown on 15 machines

3) The mathematics of finite-source model can be complex, analysts often use finite queuing tables in conjunction with simple formulas to analyze these systems

Important Note: To solve queuing problems use the Excel templates that accompany the text

Five Typical Measures of System Performance

Operations Managers Look at

1) Average number of customers waiting (in line or in system)

2) Average time customers wait (in line or system)

3) System utilization (percentage of capacity used)

4) Implied cost of given level of capacity and its related waiting line

5) The probability that an arrival will have to wait for service

Infinite-source Symbols

[pic]Problem 1 (808)

Repair calls are handled by one repairman at a photocopy shop. Repair time, including travel time, is exponentially distributed, with a mean of two hours per call. Requests for copier repairs come in at a mean rate of three per 8-hour day (assume Poisson).

Determine:

a) The average number of customers awaiting repairs.

b) System utilization

c) The amount of time during an 8-hour day that the repairman is not out on call

d) The probability of two or more customers in the system.

Excel Solution

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[pic]Problem 2 (808)

A vending machine dispenses hot chocolate or coffee. Service time is 30 seconds per cup and is constant. Customers arrive at a mean rate of 80 per hour, and this rate is Poisson distributed. Determine:

a) The average number of customer waiting in line

b) The average time customers spend in the system

c) The average number in the system

Excel Solution

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[pic]Problem 4 (809)

A small town with one hospital has two ambulances to supply ambulance service. Requests for ambulances during non-holiday weekends average 0.8 per hour and tend to be Poisson distributed. Travel and assistance time averages one hour per call and follows an exponential distribution. Find:

a) System utilization

b) The average number of customers waiting

c) The average time customers wait for an ambulance

d) The probability that both ambulances will be busy when a call comes in

Excel Solution

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Problem 10 (810)

Two operators handle adjustments for a group of 10 machines. Adjustment time is exponentially distributed and has a mean of 14 minutes per machine. The machines operate for an average of 86 minutes between adjustments. While running, each machine can turn out 50 pieces per hour. Find:

a) The probability that a machine will have to wait for an adjustment

b) The average number of machines waiting for adjustment

c) The average number of machines being serviced

d) The expected hourly output of each machine, taking adjustments into account

e) Machine downtime represents a cost of $70 per hour; operator cost (including salary and fringe benefits) is $15 per hour. What is the optimum number of operators?

Excel Solution

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