Question

A barbershop has two barbers: an experienced owner and an apprentice. The owner cuts hair at the rate of 4 customers/hour, while the apprentice can only do 2 customers/hour. The owner and the apprentice work simultaneously, however any new customer will always go rst to the owner, if the latter is available. The barbershop has waiting room for only 1 customer (in case both barbers are busy), any additional customers are turned away. Suppose customers walk by the barbershop at the rate of 6 customers/hour. Construct a model to find the proportion of time the apprentice is busy cutting hair.

Construct a continuous-time Markov chain for this problem and explain your assumptions.

Write down the in nitesimal generator G of this chain.

Using your model and the proportion of time the apprentice is busy cutting hair.

Answer 1

A continuous-time Markov chain model can be created to determine the proportion of time an apprentice barber is busy. This involves calculating the stationary distribution of the system's states based on customer arrival and service completion rates.

Step-by-step explanation:

Constructing a Continuous-Time Markov Chain Model

To find the proportion of time the apprentice barber is busy cutting hair, we can construct a continuous-time Markov chain model. Assuming customers arrive at a rate of 6 customers/hour, on average one customer arrives every ten minutes. The experienced owner cuts hair at a rate of 4 customers/hour, and the apprentice at a rate of 2 customers/hour. The owner always serves first, if available. There's only space for one waiting customer; extras are turned away. We can define the states of the Markov chain as follows:

State 0: Both barbers are free.

State 1: Owner is busy; apprentice is free. One customer is being serviced.

State 2: Both barbers are busy; no customers are waiting.

State 3: Both barbers are busy; one customer is waiting.

The transitions between these states occur based on customer arrivals and service completions. The infinitesimal generator matrix G, given the arrival and service rates, can be derived. With G, we can solve for the stationary distribution to determine the long-run proportion of time the system is in each state, particularly the proportion of time the apprentice is actively cutting hair (busy), which corresponds to States 2 and 3.