The Z-Transform Applied to Birth-Death Markov Processes

Abstract

Birth-death Markov models have been widely used in the study of natural and physical processes. The analysis of such processes, however, is mostly performed using time series analysis. In this report, a finite state birth----death Markov process is analyzed using the z----transform approach. The performance metrics of the system and their variation with the system parameters are then derived and presented.

The Z-Transform Applied to a birth-death Process Having Varying birth and death rates

“…As expected, both L nm and L mn increase with m, the implication here being that as m approaches N, it becomes more difficult for the system to transit from state 1 to m (L nm ) as well as from state m to 1 (L mn ). Results for cases where the transition probabilities do not change from state to state have been presented in (Kundaeli, 2008) where a similar trend was observed. Moreover, and although not shown, the values of L nm and L mn in the current analysis were found to be lower than those presented in the previous report.…”

“…In a previous report, the z-transform was used to analyze a birth-death process in which the birth and death transition probabilities were the same in all states (Kundaeli, 2008). In this report, we extend on the results of that report to analyze a birth-death process in which the birth and death transition probabilities can vary from state to state.…”

“…Following the approach in (Kundaeli, 2008), state reduction techniques can be used to reduce the transition diagram to that of Fig. 2 where we have used the notations M = mn, K = n and J = Nm + 1.…”

“…This is expected because the transition probabilities (P i and Q i ) in that report were fixed, whereas in this report they increase with the state. As explained in (Kundaeli, 2008), the presented model can be used to study communication and queuing systems having birth-death like structures. The significance of the present analysis is the possibility to investigate real life population systems where birth and death rates are proportional to population size, and also conditions may exist that favour births to deaths and vice versa.…”

Throughput Analysis of Arq Schemes Using State Transition Diagrams

“…An adequate review of the methods employed for each type of frame synchronized systems is beyond the scope of this report and interested readers are therefore advised to consult (Kundaeli, 1998;2007) and references therein. A good treatment of applying the ztransform to state transition diagrams is also well treated in Kundaeli (2008). In the current analysis the method of using the synchronization efficiency is applied to ARQ systems to obtain the throughput efficiency instead of the delay times.…”