M/M/C Queue Analysis

The question concerns an M/M/C queueing model with $C=3$ servers, arrival rate $\lambda$, and service rate $\mu$. We evaluate each statement regarding the system state $X(t)$, the number of customers at time $t$.

Statement Evaluation

• Statement 1: $\{X(t)\}$ is a birth and death process with constant birth and death rates.
  • The process $\{X(t)\}$ is a birth-death process.
  • The birth rate (arrival rate) is constant: $\lambda$.
  • The death rate (service completion rate) depends on the number of customers: $X(t)\mu$ if $X(t) < C$, and $C\mu$ if $X(t) \ge C$.
  • Since the death rate is not constant, this statement is false.
• Statement 2: If $\{X(t)\}$ has a stationary distribution, then $\lambda < 3\mu$.
  • A stationary distribution exists for an M/M/C queue if the traffic intensity $\rho = \frac{\lambda}{C\mu}$ is less than 1.
  • For $C=3$, the condition is $\rho = \frac{\lambda}{3\mu} < 1$.
  • This inequality simplifies to $\lambda < 3\mu$.
  • Therefore, this statement is true.
• Statement 3: If $\lambda < 3\mu$, then the stationary distribution is a geometric distribution with parameter $\frac{\lambda}{3\mu}$.
  • The stationary distribution for an M/M/C queue ($C>1$) is not geometric.
  • The probabilities $P_n$ depend on $(\lambda/\mu)^n$ and factorials, unlike a geometric distribution $P_n = (1-\rho)\rho^n$.
  • Thus, this statement is false.
• Statement 4: The number of customers undergoing service at time $t$ is $\min\{X(t), 3\}$.
  • There are $C=3$ servers available.
  • If the total number of customers $X(t)$ is less than $C$, all $X(t)$ customers are in service (or waiting to start). Assuming $X(t)$ counts those in service or waiting, the number actually *in service* cannot exceed the number of customers present. Thus, if $X(t) < 3$, $X(t)$ customers are in service.
  • If $X(t) \ge C=3$, all 3 servers are busy, and 3 customers are undergoing service.
  • Combining these cases, the number of customers in service is $\min\{X(t), 3\}$.
  • This statement is true.

Conclusion

Based on the analysis, statements 2 and 4 are true.