Question

During tax season the IRS hires seasonal workers to help answer the questions of taxpayers who call a special toll-free number for information. Suppose that calls to this line occur at an average rate of 60 calls per hour and follow a Poisson distribution. An IRS worker can handle an average of 5 calls per hour with service times, exponentially distributes. Assume that there are 10 IRS workers and when they are all busy, the phone system can keep 5 additional callers on hold.

IRS TOLL-FREE

QUEUE 1 : M / M / C / K

Q U E U E   S T A T I S T I C S (time units = hrs)

      Number of identical servers . . . . . . . . .          10

      Mean arrival rate . . . . . . . . . . . . . .               60.0000

      Mean service rate per server . . . . . . . .          5.0000

      Waiting room capacity . . . . . . . . . . . .           5

      Mean server utilization (%) . . . . . . . . .                 95.6087

      Expected number of customers in queue . . . .         2.4506

      Expected number of customers in system . . .    12.0114

      Probability that a customer must wait . . . .         0.8111

      Probability of service denial . . . . . . . .                    0.2033

                              IRS TOLL-FREE

                         QUEUE 1 : M / M / C / K

              PROBABILITY DISTRIBUTION OF NUMBER IN SYSTEM

    Number Prob 0   0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9   1

+----+----+----+----+----+----+----+----+----+----+

   UNDER    0.0059|+                                                |

         5 0.0099|+                                                |

         6 0.0199|*-                                               |

         7 0.0340|**-                                              |

         8 0.0511|***---                                           |

         9 0.0681|***+-----                                        |

        10 0.0817|****+---------                                   |

        11 0.0980|*****-------------                               |

        12 0.1176|******------------------                         |

        13 0.1412|*******+-----------------------                  |

        14 0.1694|********+-------------------------------         |

        15 0.2032|**********+--------------------------------------|

                  +----+----+----+----+----+----+----+----+----+----+

Using the computer output above respond to the following questions:

1. What percentage of the time is an IRS worker busy?

2. What percentage of callers gets a busy signal when they place their calls?

3. What is the average number of callers waiting for a turn to talk to the IRS workers?

4. What is the probability that a customer must wait?

5. What is the probability that a customer gets immediate attention and does not have to wait?

6. The arrival rate of calls is 60 calls per hour. If the office is open 8 hours a day, what is the expected number of customers denied service per day?

7. The average revenue per customer is $120. Estimate lost revenue per day due to service denial.

Answer 1