Model Setup
The system can be approximated using by a Continuous Time Markov Chain.
Let the call incoming rate be RI and average call service time be ST
Assume that there are 2 Gateways: GW1 and GW2. Let the capacities be N1 and N2 repectively.
Let's assume that the last known update had gatew states as I1 and I2. And the gateway selection probabilities for the two gateways be P1 and P2.
It is obvious that P1 + P2 = 1
This can be modeled as a CTMC being split into two CTMCs with CallIncomingRates as P1*RI and P2*RI respectively. The overall blocking probability can be computed as:
Pb = P1*Pb1 + P2*Pb2
Where Pb1 and Pb2 are the blocking probabilities for the two chains for time T - the Update Interval.
Why is steady state analysis not enough?
The steady state blocking probability for a CTMC is given by ErlangB formula. According to which the blocking probability is independent of intitial state and Update Interval. It depends only on the number of states in the chain Ni and the load Ei = Pi * RI * ST. Hence, no fruitful relationship between T and Pb can be found.
Uniformization
Uniformization is a method for transient state analysis of CTMC. The basics lay in re-conceptualizing a CTMCs by DTMCs. A very simple procedure for modeling this is given in http://www.cse.cuhk.edu.hk/~cslui/CSC5420/transient_beamer.pdf and implemented with minimal calculations [Code mailed to the Guide].
The system can be approximated using by a Continuous Time Markov Chain.
Let the call incoming rate be RI and average call service time be ST
Assume that there are 2 Gateways: GW1 and GW2. Let the capacities be N1 and N2 repectively.
Let's assume that the last known update had gatew states as I1 and I2. And the gateway selection probabilities for the two gateways be P1 and P2.
It is obvious that P1 + P2 = 1
This can be modeled as a CTMC being split into two CTMCs with CallIncomingRates as P1*RI and P2*RI respectively. The overall blocking probability can be computed as:
Pb = P1*Pb1 + P2*Pb2
Where Pb1 and Pb2 are the blocking probabilities for the two chains for time T - the Update Interval.
Why is steady state analysis not enough?
The steady state blocking probability for a CTMC is given by ErlangB formula. According to which the blocking probability is independent of intitial state and Update Interval. It depends only on the number of states in the chain Ni and the load Ei = Pi * RI * ST. Hence, no fruitful relationship between T and Pb can be found.
Uniformization
Uniformization is a method for transient state analysis of CTMC. The basics lay in re-conceptualizing a CTMCs by DTMCs. A very simple procedure for modeling this is given in http://www.cse.cuhk.edu.hk/~cslui/CSC5420/transient_beamer.pdf and implemented with minimal calculations [Code mailed to the Guide].
