This document defines key terms and equations used in queuing theory. It covers models for random arrivals with deterministic service (M/D/1), random arrivals and service with one server (M/M/1), random arrivals and service with multiple servers (M/M/C), and a maximum queue size (M/M/C/K). Key metrics defined include arrival rate, service rate, utilization, expected queue length, time in system, and wait time. Equations are provided for calculating these metrics based on the number of servers and arrival/service distributions.

1. Queuing Theory Equations
Definition
λ = Arrival Rate
μ = Service Rate
ρ = λ / μ
C = Number of Service Channels
M = Random Arrival/Service rate (Poisson)
D = Deterministic Service Rate (Constant rate)
M/D/1 case (random Arrival, Deterministic service, and one service channel)
Expected average queue length E(m)= (2ρ- ρ2
)/ 2 (1- ρ)
Expected average total time E(v) = 2- ρ / 2 μ (1- ρ)
Expected average waiting time E(w) = ρ / 2 μ (1- ρ)
M/M/1 case (Random Arrival, Random Service, and one service channel)
The probability of having zero vehicles in the systems Po = 1 - ρ
The probability of having n vehicles in the systems Pn = ρn
Po
Expected average queue length E(m)= ρ / (1- ρ)
Expected average total time E(v) = ρ / λ (1- ρ)
Expected average waiting time E(w) = E(v) – 1/μ

2. M/M/C case (Random Arrival, Random Service, and C service channel)
Note :
c
ρ
must be < 1.0
The probability of having zero vehicles in the systems
Po =
( )
1_
1
0 /1!!
⎥
⎦
⎤
⎢
⎣
⎡
−
+∑
−
=
c
n
Cn
ccn ρ
ρρ
The probability of having n vehicles in the systems
Pn = Po
!n
n
ρ
for n < c
Pn =Po
!cc cn
n
−
ρ
for n > c
Expected average queue length
E(m)=
( )2
1
/1
1
! ccc
P
c
o
ρ
ρ
−
+
Expected average number in the systems
E(n) = E(m) + ρ
Expected average total time E(v) = E(n) / λ
Expected average waiting time E(w) = E(v) – 1/μ

3. M/M/C/K case (Random Arrival, Random Service, and C service Channels and K
maximum number of vehicles in the system)
The probability of having zero vehicles in the systems
For 1≠
c
ρ
1
1
0
1
1
1
!!
1
−
−
=
+−
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
= ∑
c
n
cK
c
n
o
c
c
cn
P
ρ
ρ
ρ
ρ
For 1=
c
ρ
( )
1
1
0
1
!!
1
−
−
=
⎥
⎦
⎤
⎢
⎣
⎡
+−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
= ∑
c
n
c
n
o cK
cn
P
ρ
ρ
cn0for
!
1
≤≤= o
n
n P
n
P ρ
kncforP
!c
1
o
n
c-n
≤≤⎟
⎠
⎞
⎜
⎝
⎛
= ρ
c
Pn
( )
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+−⎟
⎠
⎞
⎜
⎝
⎛
−−⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
=
−+− ckck
c
o
c
ck
cc
c
c
c
P
mE
ρρρ
ρ
ρ
ρ
111
1!
)(
1
2
∑
−
=
−
−+=
1
0 !
)(
)()(
c
n
n
o
n
nc
PcmEnE
ρ
( )KP
nE
vE
−
=
1
)(
)(
λ
μ
1
)()( −= vEwE