customer arrive at a two server system at poisson rate 3 per hour.an arrival finding the system empty is equally likely to enter service with either server.an arrival finding one cusomer in the system will enter service with the idele server.an arrival finding two others in the system will wait in the line for the first free server.an arrival finding three in the system will not enter.all service times are exponential with rate one per hour,and once a customer is served(by either one) he will departs the system.
A) what is the average number of customers in the station?
B)what is the average amount of time that an entire customer spends in the station?
To solve this problem, we can use the concepts of queuing theory. Let's break down the problem step by step.
Step 1: Define the system
In this case, the system consists of two servers where customers can arrive and be served.
Step 2: Determine the arrival rate and service rate
The arrival rate is given as 3 customers per hour, which follows a Poisson distribution. The service rate is exponential with a rate of one customer per hour.
Step 3: Calculate the average number of customers in the station (A)
To calculate the average number of customers in the station, we need to find the average number of customers in each possible state.
State 0: The system is empty.
Since an arrival finding the system empty is equally likely to enter service with either server, the probability of entering service with any server is 1/2. Therefore, the average number of customers in this state is 0 * (1/2) = 0.
State 1: One customer is in service.
Since an arrival finding one customer in the system will enter service with the idle server, there is a 1/2 chance of finding the system in this state. Therefore, the average number of customers in this state is 1 * (1/2) = 1/2.
State 2: Two customers are in service.
Since an arrival finding two other customers in the system will wait in line for the first free server, there is a 1/2 chance of finding the system in this state. Therefore, the average number of customers in this state is 2 * (1/2) = 1.
State 3: The system is full.
Since an arrival finding three customers in the system will not enter, there is a 0 chance of finding the system in this state. Therefore, the average number of customers in this state is 3 * 0 = 0.
Average number of customers in the station (A) = average number of customers in all states = (0 + 1/2 + 1 + 0) = 3/2.
Therefore, the average number of customers in the station is 3/2.
Step 4: Calculate the average amount of time that an entire customer spends in the station (B)
To calculate the average time spent in the station, we need to consider the different scenarios:
Scenario 1: An arrival finding the system empty
In this case, the customer enters service with either server, and the average service time is 1 hour. Therefore, the average time spent in the station for this scenario is 1 hour.
Scenario 2: An arrival finding one customer in service
In this case, the arriving customer joins the server already in service, and the average remaining service time is 1/2 hour (since the rate of service is 1 customer per hour). Therefore, the average time spent in the station for this scenario is 1/2 hour.
Scenario 3: An arrival finding two customers in service
In this case, the arriving customer has to wait in line for the first free server. Since the arrival rate is 3 customers per hour, one customer arriving every 1/3 hour, the average waiting time is 1/3 hour. After waiting, the average remaining service time is 1/2 hour. Therefore, the average time spent in the station for this scenario is (1/3 + 1/2) = 5/6 hour.
Scenario 4: An arrival finding three customers in the station
In this case, the arriving customer does not enter the system, so the average time spent in the station for this scenario is 0 hours.
Now, we need to determine the probabilities for each scenario.
Probability of scenario 1: P1 = 1/2
Probability of scenario 2: P2 = 1/2
Probability of scenario 3: P3 = 1/2
Probability of scenario 4: P4 = 0
Average time spent in the station (B) = (P1 * 1) + (P2 * 1/2) + (P3 * 5/6) + (P4 * 0)
= (1/2 * 1) + (1/2 * 1/2) + (1/2 * 5/6) + (0 * 0)
= 1/2 + 1/4 + 5/12
= 6/12 + 3/12 + 5/12
= 14/12
= 7/6
Therefore, the average amount of time that an entire customer spends in the station is 7/6 hours.