Assume that the number of mobile hosts in a cell area is independent of
whether its CBR source is on or off. We can now modify the probability
that *i* sources are on and the queue fill is less than *x* by incorporating
the probability that there are at least *i* sources,
shown in Equation 8.

is the probability that at time *t*, *i* sources are on, and
the number of items in the buffer does not exceed *x*. is the
probability that there are at least *i* sources sending data to the buffer.
can be found from Equation 6 as follows,

The buffer fill distribution as defined in [1] is

Now that the channel equilibrium probabilities have been determined, we can account for the fact that the sources are mobile. Since the channel equilibrium probabilities have no dependence on time, the method of solution in [1] can be used with minor modifications,

From [1], is the equilibrium probability that *i*
sources are on, and the buffer content does not exceed *x*. Thus
is the probability that *i* out of *N* sources are
simultaneously on. In the mobile environment, this is now,

The mobile extension from Equation 8 carries through [1] for example, equation (13) in [1] is now,

and

is the right eigenvector of

where D and M are matrices used to represent the differential equation in Equation 10.

is the generating function of . These values are useful in [1] for solving the equilibrium buffer fill differential equation. The remainder of the solution is straight forward from [1]. Thus it has been shown how an analysis of constant bit rate on-off sources which model fixed length ATM packet sources, is extended to a mobile environment.

Note that the analysis uses a technique which is more accurate than
*M*/*D*/1 for the fixed size ATM cells, yet uses a memoryless analysis
for the channel holding time distribution. This is a reasonable approach
since the variable length channel hold times can be accurately modeled
by a memoryless analysis, while the *M*/*D*/1 analysis yields optimistic
results which can be replaced by the more accurate method in [1]
as this section has described.

Sun Aug 25 17:38:45 CDT 1996