Question

In: Statistics and Probability

Pete's Market is a small local grocery store with only one checkout counter. Assume that shoppers...

Pete's Market is a small local grocery store with only one checkout counter. Assume that shoppers arrive at the checkout lane according to a Poisson probability distribution, with an arrival rate of 16 customers per hour. The checkout service times follow an exponential probability distribution, with a service rate of 20 customers per hour. The manager’s service goal is to limit the waiting time prior to beginning the checkout process to no more than five minutes. Also the manager of Pete's Market wants to consider one of the following alternatives for improving service. Calculate the value of Wq for each alternative.

  1. Hire a second person to bag the groceries while the cash register operator is entering the cost data and collecting money from the customer. With this improved single-server operation, the service rate could be increased to 32 customers per hour. Round your answer to three decimal places. Do not round intermediate calculations.

    Wq ___________ = minutes
  2. Hire a second person to operate a second checkout counter. The two-server operation would have a service rate of 20 customers per hour for each server. Note: Use P0 values from Table 11.4 to answer this question. Round your answer to three decimal places. Do not round intermediate calculations.

    Wq = ___________ minutes

Solutions

Expert Solution

a.

Arrival rate = 16 customers per hour

Service rate = 32 customers per hour

average utilization of the system = p = / = 16/32 = 0.5

Wq = p/( - ) = 0.5/ (32 - 16) hour = 0.03125 hour = 1.875 minutes

b.

Arrival rate = 16 customers per hour

Service rate = 20 customers per hour

number of servers in the system, s = 2

average utilization of the system = p = /s = 16/(2 * 20) = 0.4

For s = 2

P0 = 0.4285715

Lq = P0 (/)s p / s! (1-p)2

= 0.4285715 * (16/20)2 * 0.4 / (2! * (1-0.4)2) = 0.152381

Wq = Lq / = 0.152381 / 16 hour = 0.009523812 hour = 0.5714287 minutes

orchestra answered 2 years ago
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