Final answer:
The question involves estimating the average system time for order readiness using queueing theory principles, specifically the Pollaczek–Khinchine formula. It requires the average inter-arrival and processing times, where the only information provided indicates an average inter-arrival time of two minutes for customers. Additional data would be needed to provide a specific answer.
Step-by-step explanation:
The question pertains to the estimation of the average time it takes to get an order ready for pickup at a store, considering both the waiting time before processing and the processing time itself. The Pollaczek–Khinchine formula is employed in queuing theory to determine this average system time, given the average inter-arrival times and processing times.
In the context provided, the average inter-arrival time between two successive orders is two minutes, as deduced from the given rate of 30 customers per hour. Consequently, the average time for three customers to arrive when the store opens is six minutes.
The mention of the exponential distribution of the waiting time between events, such as customer arrivals, is crucial because it is a common assumption in queueing models. The standard deviation of waiting times at different supermarkets and the post office example provided help illustrate variance in customer waiting times.
However, without the specific values for the average processing time (service rate) or the number of customers (arrival rate) specific to this scenario, the Pollaczek–Khinchine formula cannot be applied without additional data. If we had these values, we would apply the formula to find the average system time for an order to be completed with 11 in-store shoppers.