Question

Consider a train station to which customers arrive in accordance with a Poisson process having rate \(\lambda\). A train is summoned whenever there are \(N\) customers waiting in the station, but it takes \(K\) units of time for the train to arrive at the station. When it arrives, it picks up all waiting customers. Assuming that the train station incurs a cost at a rate of \(n c\) per unit time whenever there are \(n\) customers present, find the long-run average cost.

Short answer

The long-run average cost of the train station can be found by interpreting it as an M/G/1 queue system. Considering the arrival rate λ, service rate μ = N/K, and using Little's Law, we have L = λK, where L is the average number of customers in the system. The long-run average cost per unit time is then C = L*c = λKc.

Step-by-step solution

Analyze the problem as an M/G/1 queue system

The problem can be interpreted as an M/G/1 queue, where Poisson arrivals are served in batches whenever the train arrives (every K units of time). A train is summoned with N customers in the system, meaning there needs to be N-1 customers when the train is arriving.

Determine the arrival rate and service rate

The arrival rate is given, and it is λ. The service rate can be calculated as μ = N/K, as a train takes K units of time to arrive and picks up all N waiting customers.

Calculate the utilization factor

The utilization factor, ρ, in an M/G/1 queue can be calculated as the ratio between arrival rate and service rate, which is ρ = λ/μ. In this case, ρ = λK/N.

Apply Little's Law to find average number of customers in the system

Using Little's Law, we have L = λW, where L is the average number of customers in the system and W is the average time a customer spends in the system. Since W = K, we have L = λK.

Calculate the long-run average cost per unit time

We know that there is a cost at a rate of n*c per unit time for n customers present in the system. The long-run average cost per unit time, denoted by C, can be calculated as C = L*c, where L is the average number of customers in the system and c is the cost per customer per unit time. In this case, C = λKc. Thus, the long-run average cost of the train station is λKc per unit time.