An Example of M/M/1 Queue
‰ An airport runway for arrivals only
‰ Arriving aircraft join a single queue for the runway
‰ Exponentially distributed service time with a rate
µ = 27 arrivals / hour (As you computed in PS1.)
‰ Poisson arrivals with a rate λ = 20 arrivals / hour
‰ W = 1 = 1 = 1 hour ≈ 8.6 min
µ − λ 27 − 20 7
‰ L = λW = λ = 20 ≈ 2.9 aircrafts
µ − λ 27 − 20
‰ Wq = W − 1 = 1 − 1 = 1 − 1 ≈ 6.4 min
µ µ − λ µ 27 − 20 27
2
2
20
λ
‰ Lq = λWq =
=
≈ 2.1 aircrafts
µ (µ − λ ) 27(27 − 20)
1.225, 11/20/02
Lecture 8, Page 1
An Example of M/M/1 Queue (cont.)
‰ Now suppose we are in holidays and the arrival rate increases
λ = 25 arrivals / hour
‰ How will the quantities of the queueing system change?
‰ W = 1 = 1 = 1 hour = 30 min
µ − λ 27 − 25 2
λ
25
=
= 12.5 aircrafts
‰ L = λW =
µ − λ 27 − 25
‰ Wq = W −
1
µ
=
1
1
1
1
− =
−
= 27.8 min
µ − λ µ 27 − 25 27
2
252
λ
‰ Lq = λWq =
=
≈ 11.6 aircrafts
µ (µ − λ ) 27(27 − 25)
1.225, 11/20/02
Lecture 8, Page 2
An Example of M/M/1 Queue (cont.)
‰ Now suppose we have a bad weather and the service rate decreases
µ = 22 arrivals / hour
‰ How will the quantities of the queueing system change?
‰ W = 1 = 1 = 1 hour = 30 min
µ − λ 22 − 20 2
λ
20
=
= 10 aircrafts
‰ L = λW =
µ − λ 22 − 20
‰ Wq = W −
1
µ
=
1
1
1
1
− =
−
≈ 27.3 min
µ − λ µ 22 − 20 22
2
20 2
λ
‰ Lq = λWq =
=
≈ 9.1 aircrafts
µ (µ − λ ) 22(22 − 20)
1.225, 11/20/02
Lecture 8, Page 3