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December 2nd, 2018, 04:00 PM  #1 
Newbie Joined: Dec 2018 From: Australia Posts: 1 Thanks: 0  Multiclass M/M/1 queue which can simultaneously have one customer per class
Suppose there is a queue with exponential service time $\displaystyle \frac{1}{\mu}$ which accepts customers from K classes with poisson distribution and rate $\displaystyle \lambda_k$ but if a customer from any class arrives at the queue and there is already another customer from the same queue either waiting or receiving service, leaves. I have performed simulation to find the total service time $\displaystyle S = W_q + \frac{1}{\mu}$ but failed miserably. These are what I have found out from simulations so far: 1. service time $\displaystyle T_k$ is diffrent in every class. 2. $\displaystyle \lambda_k$ changes while leaving the queue server and becomes $\displaystyle \lambda_k'=\frac{\lambda_k}{1+\lambda_k*T_k}$ beacause when one customer arrives and stays in the queue for $\displaystyle T_k$, $\displaystyle \lambda_k*T_k$ customers arrive and leave immediatly without service. So one out of every $\displaystyle 1+\lambda_k*T_k$ gets service from the queue, hence $\displaystyle \lambda_k'=\lambda_k*\frac{1}{1+\lambda_k*T_k}$. 3. $\displaystyle \sum_k (\lambda_k'T_k)=(\sum_k \lambda_k')T$ in which T is the total mean service time of all customers regardless of class. __________________________________________________ _________________________ So I found out that if we consider this queue as K M/M/1/1 queues with $\displaystyle T_k$ service times, this could lead to finding the average number of customers in the system. _**The followings are verified by simulation**_. Every class has a separate M/M/1/1 queue with service time equal to $\displaystyle T_k$ that we are looking for. $\displaystyle P_0$ is the probability of each of these queues being empty which is by M/M/1/1 standards: $\displaystyle P_0^k = \frac{1}{1+\lambda_kT_k}$ Average number of customers of K in the system is equal to $\displaystyle 1P_0^k$, because every one of them could only have one customer: $\displaystyle P_1^k=n_k=\frac{\lambda_kT_k}{1+\lambda_kT_k}$ The total number of customers in the system is the sum of $\displaystyle n_k$: $\displaystyle N=\sum_kn_k $ ***By this point these are all verified by simulations***. After this I tried to find $\displaystyle T_k$s and failed. I thought $\displaystyle T_k$ should be like this: $\displaystyle T_k=(Nn_k+1)*\frac{1}{\mu}$ or $\displaystyle T_k=(1P_1^k)(Nn_k+1)*\frac{1}{\mu}$ I thought that every customer, by entering the queue will have to wait for each one of customers from other classes be served for $\displaystyle \frac{1}{\mu}$ and then be served itself. None of the above work. Could anyone Tell me why? 

Tags 
class, customer, m or m or 1, multiclass, poisson, probabibility, queue, queueing, simultaneously, statistics, stochastic process 
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