Answer:
Number of customers sent back = λa * pb
Explanation:
In a M/M/1/b queue, the arrival rate of customers follows a Poisson distribution, while the service time for each customer follows an exponential distribution. The "b" in the notation represents the maximum number of customers allowed in the system, including both those being served and those waiting in the queue.
To derive the average work in process (WIP), we need to consider the Little's Law. Little's Law states that the average number of customers in a system (W) is equal to the average arrival rate (λ) multiplied by the average time a customer spends in the system (T). In this case, WIP refers to the average number of customers in the system at any given time.
The average arrival rate (λ) can be calculated using the arrival rate (λa) and the blocking probability (pb) as follows:
λ = λa * (1 - pb)
The average time a customer spends in the system (T) is equal to the sum of the average time spent in the queue (Tq) and the average service time (Ts).
To calculate Tq, we can use Little's Law again. The average number of customers in the queue (Q) is equal to the arrival rate (λ) multiplied by the average time spent in the queue (Tq).
Q = λ * Tq
Since the system has a maximum capacity of "b" customers, the average number of customers in the system (W) is equal to the sum of the average number of customers in the queue (Q) and the average number of customers being served (S).
W = Q + S
Now, let's calculate the average length of the queue. As mentioned earlier, the average number of customers in the queue (Q) is equal to the arrival rate (λ) multiplied by the average time spent in the queue (Tq).
Q = λ * Tq
Finally, to determine the number of customers on average that this manufacturing system would send back every day due to its blocking character, we need to consider the blocking probability (pb). The blocking probability is the probability that a customer is blocked or rejected due to the system being at maximum capacity. The number of customers sent back can be calculated by multiplying the arrival rate (λa) by the blocking probability (pb).
Number of customers sent back = λa * pb