Answer to Question #255764 in Statistics and Probability for orko

Question

Draw the state transition rate diagram for the following queuing systems and from the diagram write down the equation of states: i) 𝑴/𝑴/𝟏(∞) queue with discouraged arrival rate, ii) 𝑴/𝑴/𝟏/𝑲: Finite Storage, iii) 𝑴/𝑴/𝒔/𝒔: s-Server Loss Systems, iv) 𝑴/𝑴/𝒔/𝑲/𝑴: Finite Population, s-Server Case, Finite Storage

Answer 1

i)

Interarrival times are exponentially distributed, with average arrival rate λ. Service times are exponentially distributed, with average service rate µ. There is only one server. The buffer is assumed to be infinite.

Let P_i be system in state i

We have λP_i = µP_i+1 

ii)

Similar to M/M/1, except that the queue has a finite capacity of K slots. That is, there can be at most K customers in the system. If a customer arrives when the queue is full, he/she is discarded (leaves the system and will not return).

There are no states greater than K.

$P_i=\begin{cases} \rho^iP_0 &for\ 0\le i\le K \\ 0 &for\ i>K \end{cases}$

where $\rho=\lambda/\mu$

iii)

 There are s servers and no waiting room. Calls arrive in a Poisson process with rate λ. The service time of each call has exponential distribution with mean 1/µ. Calls that arrive when all servers are busy are blocked and lost.

$k\mu P_i=\lambda P_{i-1},\ 1\le k\le s$

iv)

Finite population model: if arrival rate depends on the number of customers being served and waiting, e.g., model of one corporate jet, if it is being repaired, the repair arrival rate becomes zero.

the system has a maximum capacity, K

We consider s servers

Assuming s ≤ K, the maximum queue capacity is K – s

$P_n=\begin{cases} \frac{\lambda^n}{n!\mu^n}P_0 &for\ s=1,2,...,s \\ \frac{\lambda^n}{s^{n-s}s!\mu^n}P_0 &for\ s=s,s+1,...,K\\ 0&n>K \end{cases}$