This paper makes use of the analysis and notation in . There are two probability distributions that need to be developed: the channel holding time and the equilibrium probability of the number of channels used per base switch. The channel holding time is the probability that a particular base station's channel will be in use at a given time, or equivalently, that a particular source still exists. The equilibrium probability of channels in use for a given base switch is useful in this analysis as shown later.
The first of many simplifying assumptions is that there is a known average number of new calls per second per unit area. Let this be where R is the radius of the particular cell area. Handoffs are attempted at an average rate per cell, . The ratio of handoff attempts to new call attempts will be .
Let be the average number of new call attempts which are blocked. Then new calls are accepted at an average rate of . Similarly, let be the average number of handoff attempts which are blocked. Then handoff calls are accepted at a rate . The ratio of the average accepted handoffs to the average number of new calls accepted is . The channel holding time, , is a random variable defined as the time beginning when a channel is accessed, either via a new call or handoff, until the channel is released, via handoff or call completion. In order to define this, another random variable, is defined. is the time duration of a call, regardless of handoff or blocking. It is simplified as an exponential with average value, . Thus the pdf is
The strategy for determining the channel holding time distribution is to consider the time remaining for a call which has not been handed off yet, , and the time remaining after a handoff, . Since the call duration, is memoryless, the time remaining for a call after handoff has the same distribution as the original call duration. Let be the time the mobile unit remains in the original cell area, and be the time the mobile resides in the cell area after handoff. is the minimum of the call duration, , or the dwell time in the originating cell area, . A similar reasoning applies to the cell area into which a mobile unit has moved after a handoff; is the minimum of the call duration, , or the dwell time in the cell area after handoff, .
where is the probability that a call does not complete within the current cell area.
The distribution of channel holding time in a particular cell area is a weighted function of the equations shown in 2 above,
Substituting the values from Equation 1,
and differentiating to get the pdf,
To determine the equilibrium probability of the number of mobile hosts using a base station, approximate the channel holding time as simply an exponential distribution. The birth-death Markov chain can be used to find the equilibrium probability of the number of sources in each cell area. The up rates are and the down rates are multiples of the mean channel holding time.
Putting the Markov chain in closed form,
Note that C is the total number of channels for a base station and handoffs will fail with probability , i.e. all channels in that cell area are currently in use.