Question $5$

$(25$ points$)$ A small engine repair shop is run by two mechanics where people bring various types of equipment for repair. The shop is willing to accept all equipment that customers bring in$.$ However, unfortunately, when the customers see the floor of the shop covered with waiting jobs, they go elsewhere for more prompt service. The shop floor can house a maximum of $2$ equipment, excluding those being serviced. The customers arrive at the shop every $10$ minutes on the average, and it takes a mechanic an average of $20$ minutes to complete each job. Both the interarrival and service times are exponential. Determine the following quantities in steady$-$state:

a$.(20$ points$)$ Average number of idle mechanics.

b$.(5$ points$)$ When there is one equipment in the line, what is the probability that a customer arrives before a job is completed?