How do I solve a formula or equation for the Erlang System M/G/s/GD/s/infinity that predicts resource requirements (how many servers) using the known variables (1) new events per unit of time; (2) average time per event; (3) event time service level (must be resolved by duration); (4) percent of events that must meet that event time service level (duration).
Let us consider the queuing system for which
a(t)=rl(rlt)^(r-1)*e^(-rlt)/(r-1)!, t>=0 (1)
b(x)=ue^(-ux) x>=0 (2)
When we find k customers in this system and when arriving customer is in the ith stage of arrival(1<=i<=r) then the total number of stages of arrival in the system is given by
Let us use P(j)=Prob[j stages in system] so that P(j) is defined to be the number of arrival stages in the system. As always p_(k) will be the equilibrium probability for number of customers in the system, and clearly they are related through
The system we have defined is an irreducible ergodic Markov chain with its state-transition-rate diagram for stages given below
This shows how to solve a formula for the Erlang System M/G/s/GD/s/infinity that predicts resource requirements using given variables.