Queuing Formula: Single Server
λ
Utilization : ρ= =λτ
µ
Little 's law : L=λW
W = expected wait time in system = D +
1
= D+τ
µ
D = expected wait time in queue for G1/G/1
ρ ⎛ 1 ⎞ ⎛ SCVa + SCVs ⎞
≅
⎟
⎜ ⎟⎜
1− ρ ⎝ µ ⎠ ⎝
2
⎠
ρ
⎛ SCVa + SCVs ⎞
≅
( τ) ⎜
⎟
1− ρ ⎝
2
⎠
Copyright 2005 Stephen C. Graves
All Rights Reserved
Queuing Formula: Multiple Servers
k
k−1
⎛
⎞
⎛
⎛ 1 ⎞ ( kρ)
⎛ ρ ⎞⎛ 1 ⎞ ( kρ) ⎞
For M/M/k system: D = ⎜
⎟ π0 = ⎜ ⎟⎜ ⎟ ⎜
⎟ π0
⎟ ⎜⎜
⎝ kµ−λ ⎠ ⎝ (1−ρ) k!⎟⎠
⎝ 1-ρ ⎠⎝ µ ⎠ ⎜⎝ (1−ρ) k!⎟⎠
λ
1
where π0 =
and ρ =
k
i
k−1
kµ
( kρ) + ( kρ)
i!
(1−ρ) k! ∑
i=0
D = expected wait time in queue for M/G/k
⎛ 1 + SCVs ⎞
≅ ( expected wait time in queue for M/M/k ) ⎜
⎟
2
⎝
⎠
ρ ⎛ τ ⎞ ⎛ SCVa + SCVs ⎞
A quick approx. is
D≈
⎜ ⎟⎜
⎟
1− ρ ⎝ k ⎠ ⎝
2
⎠
Copyright 2005 Stephen C. Graves
All Rights Reserved
Queuing Formula: Multiple Servers, No Queue
(λτ )
For M /G /k/k: P r [ # in system = n ] = n !
n
k
∑
i=0
(λ τ )
i
and
i!
P r[ # in system = k ] = "loss probability"
For M/G/∞, number in the system is Poisson with mean = λτ and
with variance = λτ
Copyright 2005 Stephen C. Graves
All Rights Reserved
Cool off
Finish
line
Recovery
OK
Services
Blisters & Cramps
Not
OK
Dehydration
Triage
Emergency Care
Copyright 2005 Stephen C. Graves
All Rights Reserved
Cool off
Finish
line
Recovery
p = 0.975
Services
Blisters & Cramps
0.6
1-p =0.025
0.3
Dehydration
0.1
Emergency Care
Copyright 2005 Stephen C. Graves
All Rights Reserved
Cool off
λ = 80/min
Recovery
78/min
Services
Blisters & Cramps
1.2/m
2/min
0.6/m
Dehydration
0.2/m
Emergency Care
Copyright 2005 Stephen C. Graves
All Rights Reserved
Blisters & Cramps
τ = 15 min
Not OK
2/min
What’s the
wait for a nurse?
1.2
Triage
τ = 1 min
How many
servers?
0.6
Dehydration
τ = 60 min
How many cots
are needed?
0.2
Emergency Care
τ = 15 min
Copyright 2005 Stephen C. Graves
All Rights Reserved
What’s the prob.
need a back up?
λτ = workload per minute
Blisters & Cramps
τ = 15 min
1.2
Not OK
2/min
0.6
Triage
τ = 1 min
λτ = 2
Dehydration
τ = 60 min
0.2
Emergency Care
τ = 15 min
How many
servers?
What’s the
wait for a nurse?
λτ = 18
How many cots
are needed?
λτ = 36
What’s the prob.
need a back up?
λτ = 3
Copyright 2005 Stephen C. Graves
All Rights Reserved
λτ = workload per minute
Triage
τ = 1 min
How many
servers?
Not OK
2/min
M/G/k queue
λτ = 2
k=3
ρ = 0.66
expected wait time in queue
ρ ⎛ τ ⎞ ⎛ SCVa + SCVs ⎞
≈
⎜ ⎟⎜
⎟
1− ρ ⎝ k ⎠⎝
2
⎠
Copyright 2005 Stephen C. Graves
All Rights Reserved
λτ = workload per minute
1.2/min
Blisters & Cramps
τ = 15 min
What’s the
wait for a nurse?
M/G/k queue
λτ = 18
k > 18
ρ = λτ/k
expected wait time in queue
ρ ⎛ τ ⎞ ⎛ SCVa + SCVs ⎞
≈
⎜ ⎟⎜
⎟
1− ρ ⎝ k ⎠⎝
2
⎠
Copyright 2005 Stephen C. Graves
All Rights Reserved
λτ = workload per minute
How many cots
are needed?
0.6/min
Dehydration
τ = 60 min
Model as
M/G/k/k or
M/G/inf queue
λτ = 36
E[# runners] = 36
V[# runners] = 36
σ [# runners] = 6
Copyright 2005 Stephen C. Graves
All Rights Reserved
λτ = workload per minute
What’s the prob.
need a back up?
0.2/min
Emergency Care
τ = 15 min
Model as M/G/k/k
λτ = 3
(λ τ )
Loss probability = P r [ # in system = k ] =
k
k!
k
∑
i=0
(λ τ )
i
i!
k
1
2
3
4
5
6
7
Pr[#=k] 0.75 0.53 0.35 0.21 0.11 0.05 0.02
Copyright 2005 Stephen C. Graves
All Rights Reserved
OK
78/min
Cool off
τ = 10 min
Recovery
τ = 20 min
Copyright 2005 Stephen C. Graves
All Rights Reserved
Services
Rest rooms (5 min)
Phones (2 min)
Clothes (1 min)
Food (2 min)
OK
78/min
Cool off
τ= 10 min
λτ = 780
λτ = workload per minute
Recovery
τ= 20 min
λτ = 1560
Services
Rest rooms: λτ = 390
Phones: λτ = 156
Clothes: λτ = 78
Food: λτ = 156
Copyright 2005 Stephen C. Graves
All Rights Reserved