Operations and Supply Chain Management The Core

SERVICE PROCESSES chapter 7 229
Model 1 (See Exhibit 7.12.)
​λ 2 ​
​L q = ​ _______
μ(μ − λ) ​ ​W ​L q
q = ​ ___
λ ​
λ
​ ​L s = ​ ____
μ − λ ​ ​W s = ​
n
​P n = ​
( 1 − ​ _ λ μ ​ λ__
)( μ) ​ ​ ​P 0 = ​( 1 − ​ _ λ μ ​ ) ​
___ ​L s
λ ​ ρ = ​ __ λ ​ [7.3]
μ ​ ​
Model 2
​λ 2 ​
​L q = ​ ________
2μ(μ − λ) ​ ​W ​L q
q = ​ ___
λ ​ ​ [7.4]
λ
​L s = ​L q + ​ __
μ ​ ​W ​L s
s = ​ ___
λ ​
Model 3
λ
​L s = ​L q + ​ __
μ ​ ​W ​L s
s = ​ ___
λ ​ ​
​L q
​W q = ​ ___
λ ​
​P w = ​L q (
Sμ_
λ ​ − 1​ ) ​ [7.5]
SOLVED PROBLEMS
SOLVED PROBLEM 1
Quick Lube Inc. operates a fast lube and oil change garage. On a typical day, customers arrive at the
rate of three per hour, and lube jobs are performed at an average rate of one every 15 minutes. The
mechanics operate as a team on one car at a time.
Excel:
Assuming Poisson arrivals and exponential service, find:
Queue
a. The utilization of the lube team.
b. The average number of cars in line.
c. The average time a car waits before it is lubed.
d. The total time it takes to go through the system (that is, waiting in line plus lube time).
Solution
λ = 3, μ = 4
a. Utilization ​ρ = ​ __ λ μ ​ = ​ __ 3 ​ = 75 percent​.
4
​λ 2 ​
b. ​L q = ​ ________
μ(μ − λ) ​ = ________ ​ ​3​
2 ​
4(4 − 3) ​ = __ ​ 9
​ = 2.25 cars in line​.
4
​L q
c. ​W q = ​ ___ 2.25
​= ​ ____ ​ = 0.75​ hour, or 45 minutes.
λ 3
​L s
d. ​W s = ​ __
λ ​ = ​ _____ λ
μ − λ ​/ λ = ​ _____ 3
4 − 3 ​ / 3 = 1 hour (waiting + lube)​.