Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the workstation is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the workstation at any given point of time is ___________ (round off to the nearest integer).

The correct answer is 4.

Problem Understanding:
We are given a single-machine workstation where:
1. Job arrivals follow a Poisson process.
2. Service times (process times) at the workstation follow an exponential distribution.
This setup corresponds to a standard M/M/1 queuing system (Markovian arrival, Markovian service, single server).

Step 1: Identify and Align the Given Parameters
First, we need to express all rates in consistent units (jobs per hour).
The mean arrival rate, denoted by $\lambda$ (lambda), is given as:

$\lambda =12\text{ jobs/hour}$

The mean process (service) time per job, denoted by $t$, is:

$t=4\text{ minutes}$

To convert the service time into hours:

$t=\frac{4}{60}\text{ hours}=\frac{1}{15}\text{ hours}$

The service rate, denoted by $\mu$ (mu), is the reciprocal of the mean service time:

$\mu =\frac{1}{t}=\frac{1}{1/15}=15\text{ jobs/hour}$

Step 2: Calculate the Expected Number of Jobs at the Workstation
In an M/M/1 queue, the expected number of jobs at the workstation (both waiting in queue and being served), representing the length of the system ($$_Ls), is calculated using the formula:

$L_s=\frac{\lambda }{\mu -\lambda }$

Substitute the values of $\lambda =12$ and $\mu =15$ into the formula:

$L_s=\frac{12}{15-12}$

Simplify the denominator:

$L_s=\frac{12}{3}$

Solve the division:

$L_s=4\text{ jobs}$

Thus, the expected number of jobs at the workstation at any given point in time is 4.