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Mobile Node Analysis

This paper makes use of the analysis and notation in [4]. 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 tex2html_wrap_inline796 where R is the radius of the particular cell area. Handoffs are attempted at an average rate per cell, tex2html_wrap_inline800 . The ratio of handoff attempts to new call attempts will be tex2html_wrap_inline802 .

Let tex2html_wrap_inline804 be the average number of new call attempts which are blocked. Then new calls are accepted at an average rate of tex2html_wrap_inline806 . Similarly, let tex2html_wrap_inline808 be the average number of handoff attempts which are blocked. Then handoff calls are accepted at a rate tex2html_wrap_inline810 . The ratio of the average accepted handoffs to the average number of new calls accepted is tex2html_wrap_inline812 . The channel holding time, tex2html_wrap_inline814 , 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, tex2html_wrap_inline816 is defined. tex2html_wrap_inline816 is the time duration of a call, regardless of handoff or blocking. It is simplified as an exponential with average value, tex2html_wrap_inline820 . Thus the pdf is

  equation196

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, tex2html_wrap_inline822 , and the time remaining after a handoff, tex2html_wrap_inline824 . Since the call duration, tex2html_wrap_inline816 is memoryless, the time remaining for a call after handoff has the same distribution as the original call duration. Let tex2html_wrap_inline828 be the time the mobile unit remains in the original cell area, and tex2html_wrap_inline814 be the time the mobile resides in the cell area after handoff. tex2html_wrap_inline822 is the minimum of the call duration, tex2html_wrap_inline816 , or the dwell time in the originating cell area, tex2html_wrap_inline828 . A similar reasoning applies to the cell area into which a mobile unit has moved after a handoff; tex2html_wrap_inline824 is the minimum of the call duration, tex2html_wrap_inline816 , or the dwell time in the cell area after handoff, tex2html_wrap_inline814 .

Thus,

  eqnarray209

where tex2html_wrap_inline844 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,

equation218

Substituting the values from Equation 1,

  equation225

and differentiating to get the pdf,

  eqnarray229

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 tex2html_wrap_inline846 and the down rates are multiples of the mean channel holding time.

Putting the Markov chain in closed form,

  equation242

where,

equation245

Note that C is the total number of channels for a base station and handoffs will fail with probability tex2html_wrap_inline850 , i.e. all channels in that cell area are currently in use.


next up previous
Next: Mobile CBR Source Analysis Up: Mobile Systems Analysis Previous: Mobile Systems Analysis

Steve Bush
Sun Aug 25 17:38:45 CDT 1996