Useful Formulas
1. If
T
is an exponentially distributed random variable with mean
1
/λ
, then the density of
T
is
given by
f
T
, where:
f
T
(
t
) =
λe

λt
,
t
≥
0
.
2. If
S
is a random variable with a gamma distribution of parameters
n
and
λ
, where
n
is a
positive integer, then the density of
S
is given by
f
S
, where:
f
S
(
s
) =
λe

λs
(
λs
)
n

1
(
n

1)!
,
s
≥
0
.
3. For an
M/M/
1
queue with arrival rate
λ
and service rate
μ
with
λ < μ
, the equilibrium
distribution is:
P
(
Q
=
j
) = (1

ρ
)
ρ
j
,
j
= 0
,
1
,
2
, . . . ,
where
ρ
=
λ/μ
.
4. For an
M/M/
∞
queue with arrival rate
λ
and service rate
μ
, the equilibrium distribution is:
P
(
Q
=
j
) =
e

ρ
ρ
j
j
!
,
j
= 0
,
1
,
2
, . . . ,
where
ρ
=
λ/μ
.
5. For an
M/M/K/K
queue with arrival rate
λ
and service rate
μ
, the equilibrium distribution
is:
P
(
Q
=
j
) =
ρ
j
/j
!
1 +
···
+
ρ
K
/K
!
,
j
= 0
,
1
,
2
, . . . , K,
where
ρ
=
λ/μ
.
2