Final answer:
To find the probability that a truck has to wait at least 5 minutes, we need to calculate the traffic intensity and then use the exponential distribution formula for wait times in an M/M/s queue, considering both the arrival rate, service rate, and the number of service channels (docks).
Step-by-step explanation:
We're given that trucks arrive at a warehouse according to a Poisson distribution at an average rate of 6.6 trucks per hour, and the loading of trucks follows an exponential distribution at a rate of 1.2 trucks per hour. We want to determine the probability that a truck has to wait at least 5 minutes in the queue.
To find this probability, we need to consider the arrival and service rates. The arrival rate λ (lambda) is 6.6 trucks per hour and the service rate μ (mu) is 1.2 trucks per hour per dock. Since there are 7 docks, the total service rate is 7 × 1.2 = 8.4 trucks per hour.
With the arrival and service rates, we can determine the traffic intensity Ρ (rho) which is λ / μ; in this case, 6.6 / 8.4. Since Ρ is less than 1, the system is stable and trucks can be serviced without the queue becoming infinitely long.
We are interested in the probability that a truck has to wait at least 5 minutes, which is the same as saying the queue length is at least 1. This probability can be calculated using the formula for the wait times in an M/M/s queue (where 'M' represents memoryless, and 's' the number of service channels, in this case, 7):
P(wait >= t) = 1 - e^(-μ*(1-Ρ)*t)
Since we are interested in a 5 minute wait time, and knowing that our units are in hours, we convert 5 minutes to hours (5/60). Substituting the values into the formula, we can find the required probability.