50 QUEUES AND PERFORMANCE ANALYSIS

If we specialize the formula to the GI /GI /1 case, then we see E

A

i

[W

0−

σ

i,0

]=ρ

i

E

A

i

[W

0−

],

and hence we obtain that

M

i=1

ρ

i

(E

A

i

[W

0−

]+

1

2

E

A

i

[R

i

]) = E[W

0

]

where R

i

is the stationary residual time of σ

i,0

.

When specialized to the M/G/1 case we see E

A

i

[W

0−

]=E[W

0

] by PASTA, and then we

see that:

E[W

0

]=

M

i=1

ρ

i

E[R

i

]

2(1 −

M

i=1

ρ

i

)

which is just the P-K formula obtained for a M/G/1 queue with an input as a Poisson process with

rate

M

i=1

λ

i

, and the work brought has the distribution σ = σ

i

with probability

λ

i

M

i=1

λ

i

. Moreover,

it is also the mean workload seen by each type of arrival.

With this, we conclude the discussion of the main formulae associated with mean queue

lengths and waiting times.

2.4 FROM MEANS TO DISTRIBUTIONS

So far, we have concentrated on the computation of means. However, the RCL also turns out to be a

very effective tool to compute distributions of queues which we turn our attention to in this section.

2.4.1 EQUILIBRIUM DISTRIBUTIONS

Let us ﬁrst begin by computing the stationary distribution of the forward recurrence time. This is

called the equilibrium distribution of R

0

. In Chapter 1, the expression for the mean of the forward

recurrence time is to be computed under the Palm distribution. In the M/G/1 case because of

PASTA, we obtain the mean under the stationary distribution.

Let R

t

= T

+

(t) − t where by deﬁnition T

+

(t) = T

N

t

+1

where {T

n

} are the points of a sta-

tionary point process N

t

. Deﬁne: X

t

= (R

t

− x)

+

.

Then X

+

t

=−1I

[R

t

>x]

and the jumps of X

t

are given by: X

T

n

= (R

T

n

− x)

+

− (R

T

n−

− x)

+

.

By deﬁnition R

T

n

= T

n+1

− T

n

and R

T

n−

= 0, and therefore for any x ≥ 0, we have

X

T

n

= (T

n+1

− T

n

− x)

+

Applying the RCL, we obtain:

E[1I

[R

0

>x]

]=λ

N

E

N

[(S

0

− x)

+

]=λ

N

∞

x

(y − x)dF (y)

where F(x) is the distribution of the inter-arrival time S

0

= T

1

− T

0

under P

N

.

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