Problem $3$

A large pharmacy operates $10$ checkout counters each of which is staffed by a clerk. Sometimes

at a counter there is a question that needs to be resolved by an experienced roaming

supervisor. When a question arises, a signal is sent to the supervisor and the counter waits

$($and does not continue to process customers$)$ until the question is resolved. The questions

arise about once every $30$ minutes at each counter, with the times between questions being

exponentially distributed. It takes a roaming supervisor an average of $2$ minutes to resolve a

question and return the counter to service and these times are exponentially distributed as

well. A roaming supervisor makes $$55$ per hour, busy or idle. It is estimated that time that a

checkout counter spends waiting for a question to be answered costs $$75$ per hour if actually

waiting and $$65$ per hour while the question is being answered. Should the pharmacy hire a

second roaming supervisor?