Floyd\'s Barber Shop- Customers arrive and wait for Floyd the barber if there is
ID: 3134837 • Letter: F
Question
Floyd's Barber Shop- Customers arrive and wait for Floyd the barber if there is an empty seat in the waiting area. Otherwise, they go away. There arc four chairs in the waiting area. If the system is empty, Floyd takes a 15- minute break. Floyd is extremely predictable and will always take exactly 15 minutes to cut someone's hair. Let Y = the number of customers that arrive to the shop during any 15 minute interval. Assume the mass function for Y. the number of arrivals during a 15 minute interval is as follows. Develop the one-step transition probability matrix, P. Be sure to define the states and events that cause a transition, and a transition function, Given that the shop opens with no customers, what is the probability that the shop is full 1 hour later? What is the expected number of customers in the shop 1 hour later?Explanation / Answer
Answer a)
Given that the markov chain is in state i at a time t.
the probability that the Markov chain will be in state j at time t+1 is denoted by
Pij = P[x(t+1) = j | x(t) = i ]
There for Pij = P [0.4,0.2]
= P [0.25,0.4]
= P [0.14,0.25]
= P [0.01,0.14]
Answer b)
If shop opens with no customers,
in First 15 min probability of 4 customers coming is = 0.01
Floyd is taking exactly 15 min for each customer. so these 4 customers will be occupied by Shop.
in coming every 15 min interval probability of one person coming is = 0.4 * 0.4 * 0.4* 0.4 = 0.0256
Answer c)
Expected no of customers in the 1 hrs later = (0.2*0)+(0.4*1)+(0.25*2)+(0.14*3)+(0.01*4)-(1) = 0.36 same pattern will follow in coming 15 min intervals
If waiting chair will not available then customer will go away so, after 1 hr 2 customers can be available in barber shop.
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