# Queing theory

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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 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.

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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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###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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- "Thank you very much for your valuable time and assistance!"

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