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Waiting Lines. Students should be able to: Categorize a waiting line problem as to customer population, waiting line arrangement, and service facility arrangement. Describe the operating characteristics of waiting line systems and the important decisions.

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waiting lines
Waiting Lines
  • Students should be able to:
  • Categorize a waiting line problem as to customer population, waiting line arrangement, and service facility arrangement.
  • Describe the operating characteristics of waiting line systems and the important decisions.
  • Apply the single-server and multiple-server models.
  • Use Little’s Law to estimate the number of customers in the service system.
waiting line models

Customer population

Service system

Waiting line

Served customers

Service facilities

Priority rule

Figure C.1

Waiting Line Models

waiting line arrangements

Service facilities

(a) Single line

Service facilities

(b) Multiple lines

Waiting Line Arrangements

Figure C.2

operating characteristics

Operating Characteristics

  • Line length
  • Number of customers in system
  • Waiting time in line
  • Total time in system
  • Service facility utilization
decision areas

Decision Areas

  • Arrival rates
  • Number of service facilities
  • Number of phases
  • Number of servers per facility
  • Server efficiency
  • Priority rule
  • Service facility arrangement
slide6

Waiting Line Assumptions

  • Customer population : Infinite and patient
  • Customer arrivals : Poisson distribution
  • Mean arrival rate = l
  • 3. Service times : Exponential distribution
  • Mean service rate = m
  • Priority rule : First-come-first-served (FCFS)
  • Waiting line : Unlimited length
slide8

30

35

Single-Server Model

Single-Channel, Single-Phase System

Arrival rate = 30/hour Service rate = 35/hour

l

m

Utilization =  = = = 0.857, or 85.7%

m - l

30

35 – 30

l

Average number in system = L = = = 6 customers

Average number in line = Lq =  L = 0.857(6) = 5.14 customers

1

m - l

1

35 – 30

Average time in system = W =

=

=

0.20 hour

Average time in line = Wq=  W = 0.857(0.20) = 0.17 hour

Example C.3

slide10

Service facility 1

Service facility 2

Service Facility Arrangements

Multiple channel, Single phase

slide11

Multiple-Server Model

Multiple-Channel, Single-Phase System

Distribution Center

s = 4 unloading bays

λ = 3 trucks arrive per hour

μ = 1 truck unloaded per hour per bay

Suppose that it was observed that the average number of trucks waiting to be unloaded is 1.528 trucks.

hours waiting

hours being unloaded and waiting

trucks in the system

utilization

slide12

Designing a Distribution Center

Proposed design: 4 unloading bays

Employees required: 2 per bay -- $30 per hour each

Truck arrivals: 3 per hour

Unloading rate: 1 truck per hour per team

Idle truck cost: $100 per hour

Number Number of Employee Number of Truck

of BaysEmployeesCost/HrTrucks in SystemCost/HrTotal

4 8 $240 4.53 $453 $693

5 10 $300 3.35 335 635

6 12 $360 3.10 310 670

7 14 $420 3.03 303 723

5 Bays are the least expensive to operate.

choosing a service facility arrangement
Choosing a Service Facility Arrangement
  • Suppose you have two servers, each can service 5 customers per hour.
  • In Design 1, which is two single-line, single-server facilities, each server has her own line and each receives and average of 4 customers per hour.
  • In Design 2, which is a single-line, multiple-server facility, there are two servers and an average demand of 8 customers per hour.

How do the designs affect customer service?

slide14

How Does Facility Arrangement Affect Customer Service?

Design 1 Design 2

Utilization

Number in System

Number in Line

Wait Time in System

Wait Time in Line

slide15

NYPD Patrol System

  • Data
  • The average service time is 30 minutes.
  • Precinct statistics
  • Precinct Average Calls Total Felonies Size
  • per Hour per Year Sq. Miles
  • ________ ___________ ___________ _______
  • A 6.4 10,600 1.9
  • B 3.8 6,600 .8
  • If you had a total of 9 cars to allocate, how would you do it?
slide16

Precinct Waiting Statistics

Precinct A

# Cars

Utilization

4

5

6

7

8

.373

.080

.023

.007

.002

2.390

.513

.145

. 043

. 012

.800

.640

.533

.457

.400

Precinct B

# Cars

Utilization

2

3

4

5

6