Consider a simple barber shop with one (1) “service chair” and two (2) “waiting chairs.” Customers to this barber shop arrive with exponential arrival events depending on the state of the system. If the system is empty, customers arrive at rate m. If there is a customer in the chair, but no one in the waiting chairs (a total of one customer in the system), the arrival rate slows to 0.8K. If there is one customer in service and one waiting (a total of two customers in the system), the arrival rate slows further to 0.6m. Once there are three customers in the system, an incoming customer decides to turn away.
The barber also changes his work rate according to the system state. Service times are also exponential. If there is one customer in the chair and no customer waiting, the barber works at rate p. If there is one customer waiting (a total of two customers in the system), the barber speeds up to rate 1.1 g; if there are two waiting (a total of three customers in the system), he works at rate 1.2p.
If is estimated to be 3 customers per hour, and is estimated at 4 customers per hour;
a) Draw and label a state space diagram, showing all states and the rates of transition between states.
b) Write a complete set of balance equations
c) Solve your system of equations to determine the long run probability of the system being in each possible state
What is the average number ofjobs in
the system at any point in time?