# Queing theory

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).

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#### Solution Preview

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

j=rk+i-1

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

p_(k)=P(rk)+P(rk+1)+...+P(r(k+1)-1). (**)

The system we have defined is an irreducible ergodic Markov chain with its state-transition-rate diagram for stages given below

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#### Solution Summary

This shows how to solve a formula for the Erlang System M/G/s/GD/s/infinity that predicts resource requirements using given variables.