Operations and Supply Chain Management The Core

220 OPERATIONS AND SUPPLY CHAIN MANAGEMENT
exhibit 7.11
Properties of Some Specific Waiting Line Models
MODEL
LAYOUT
SERVICE
PHASE
SOURCE
POPULATION
ARRIVAL
PATTERN
QUEUE
DISCIPLINE
SERVICE
PATTERN
PERMISSIBLE
QUEUE
LENGTH
TYPICAL
EXAMPLE
1. Simple System Single channel Single Infinite Poisson FCFS Exponential Unlimited Drive-in teller at
bank; one-lane
toll bridge
2. Constant Service
Time System
3. Multichannel
System
Single channel Single Infinite Poisson FCFS Constant Unlimited Roller coaster
rides in
amusement park
Multichannel Single Infinite Poisson FCFS Exponential Unlimited Parts counter in
auto agency
exhibit 7.12
Notations for Equations
INFINITE QUEUING NOTATION: MODELS 1–3
λ = Arrival rate
μ = Service rate
1__ ​ = Average service time​
μ
1__ ​= Average time between arrivals​
λ
ρ = Ratio of total arrival rate to service rate for a single server ​
λ__
( μ)
​
L q = Average number waiting in line
L s = Average number in system (including any being served)
W q = Average time waiting in line
W s = Average total time in system (including time
to be served)
n = Number of units in the system
S = Number of identical service channels
P n = Probability of exactly n units in system
P w = Probability of waiting in line
EQUATIONS FOR SOLVING THREE MODEL PROBLEMS
Model
1
Model
2
Model
3
⎧
⎪​L q = ​ _______ ​λ 2 ​
⎨
μ(μ − λ) ​ ⎪
​
λ
L s = ​ _____
⎩ μ − λ ​
⎧
⎪​L q = ​ ________ ​λ 2 ​
⎨
2μ(μ − λ) ​ ⎪
​
λ
L s = ​L q + ​ __
⎩ μ ​
⎧
⎪​L s = ​L q + ​ __ λ
⎨
μ ​ ⎪
​
​L q
​W q = ​ __
⎩ λ ​
​L q
W q = ​ __
λ ​
​L s
W s = ​ __
λ ​
​L q
W q = ​ __
λ ​
​L s
W s = ​ __
λ ​
​L s
W s = ​ __
λ ​
P w = ​L q (
Sμ ___
λ ​− 1​ )
n
P n = ​( 1 − ​ __ λ
μ ​ __ λ
)( μ ​ ​ )
​ρ = ​
λ __
μ ​
P 0 = ​ (
1 − ​
λ __
μ ​ )
[7.3]
[7.4]
[7.5]
Problem 2: Equipment selection. A franchise for Robot Car Wash must decide
which equipment to purchase out of a choice of three. Larger units cost more but wash
cars faster. To make the decision, costs are related to revenue.