HOMEWORK 4 1. PROBLEM 1: Consider a renewal system, with cycle durations having cumulative distributed function F and density f. The spread at time t is the duration of the cycle containing t....

PROBLEM 1:
Consider a renewal system, with cycle durations having cumulative distributed function F and
density f. The spread at time t is the duration of the cycle containing t. Determine the
equilibrium density of spread.
Solution:
A renewal process is an arrival process in which the interarrival intervals are positive,
independent and identically distributed (IID) random variables (RV’s). Given that each renewal
counting processes has the same cumulative distribution function F and that a density f exists
for the cycle durations.
We know that the distribution of the interval from t to the next renewal approaches
( ) (

, -
) ∫, ( )-
( )
Where,
, - ∫ ( )
This suggests that if we look at this renewal process starting at some very large t, we should see
a delayed renewal process for which the distribution G(x) of the first renewal is equal to the
residual life distribution FY (x) above and subsequent inter-renewal intervals should have the
original distribution F(x) above. Thus it appears that such a delayed renewal process is the same
as the original ordinary renewal process, except that it starts in “steady-state.” To verify this,
we show that m(t) = t/X2 is a solution to:
( ) ( ) ∫ ( ) ( )
( )
( ) ( )
Where,
( ) , ( )-
( ) * +
( ) ( -

Substituting (t − x)/X2 for m (t − x) in equation (2):
( )
∫ , ( )-
̅̅ ̅

∫ , - ( )
̅̅ ̅

∫ , ( )-
̅̅ ̅

∫ ( )
̅̅ ̅
̅̅ ̅

Where from general renewal theorem,


( )

̅

̅̅ ̅ ∫, ( )-
Then it can be easily proved,


( )

̅̅ ̅

PROBLEM 2:
Consider a computer system that needs to implement some form of control for how jobs from
various users are allowed to access system resources. Jobs that are granted access to the
system are immediately dispatched for processing. User i generates new jobs into the system
according to a Poisson process of rate . Access control is enforced by way of permits, with
each new job requiring a permit in order to enter the system. User i start with a total number of...