GATE ME 2014 Set 4 — Question 61

The problem describes a queuing system. We are given:

• The arrival rate of jobs, $\lambda = 5$ jobs/minute.
• The mean time spent on each job (service time), $T_s = 1/8$ minute/job.

From the mean service time, we can calculate the service rate, $\mu$, which is the number of jobs that can be processed per unit time:
$\mu = \frac{1}{T_s} = \frac{1}{1/8} = 8$ jobs/minute.This scenario can be modeled as a simple M/M/1 queuing system, assuming Poisson arrivals and exponential service times, which is standard for such problems unless specified otherwise.The question asks for the mean steady state number of jobs in the system, denoted by $L_s$.The formula for the mean number of jobs in the system in an M/M/1 queue is:$$L_s = \frac{\lambda}{\mu - \lambda}$$For this formula to be valid, the system must be stable, which means the traffic intensity $\rho = \frac{\lambda}{\mu}$ must be less than 1.
In this case, $\rho = \frac{5}{8} = 0.625 < 1$, so the system is stable and a steady state exists.Now, we can substitute the values of $\lambda$ and $\mu$ into the formula for $L_s$:$$L_s = \frac{5}{8 - 5} = \frac{5}{3}$$Calculating the numerical value:$$L_s \approx 1.6667$$This value falls within the specified answer range of 1.62 to 1.70.