Queueing Analysis for an Ice Cream Vending Machine
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EXAMPLE 3: Ephebi operates an automatic ice cream vending machine at the PHC lounge. Customers arrive in a random fashion, approximated by the poisson distribution, at the rate of 45 per hour at peak demand period. The vending machine is programmed to serve customers at a constant rate of 0.8 minutes per customer. Determine: 1. The average number of customers in the queue 2. The average time spent by a customer at the vending shop.
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Hi Merry, let's solve this queuing theory problem together. First, let's read the details and identify the parameters of our system.
Given Information
* Arrival process: Poisson with rate $\lambda = 45$ customers per hour
* Service time: Constant at $0.8$ minutes per customer
Because the service time is constant, this is an M D one queuing system, where D represents deterministic or constant service times.
Model: $M/D/1$ Queue
Let's first convert the units so that the arrival rate and service rate are in the same time unit. We will use hours.
Step 1: Calculate the Service Rate ($\mu$)
The service time is zero point eight minutes per customer. In hours, that is zero point eight divided by sixty.
Multiplying one point two five customers per minute by sixty minutes gives us seventy-five customers per hour.
Now, let's calculate the system utilization factor, rho, which is lambda divided by mu.
Step 2: System Utilization ($\rho$)
Substituting forty-five for lambda and seventy-five for mu, we get zero point six.
Let's solve part one: the average number of customers in the queue, represented by L sub q.
Part (i): Average Number of Customers in Queue ($L_q$)
For an $M/D/1$ queue, the formula for $L_q$ is:
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