Finite Queue Length

[Page 587 ( continued )]

For some waiting line systems, the length of the queue may be limited by the physical area in which the queue forms; space may permit only a limited number of customers to enter the queue. Such a waiting line is referred to as a finite queue and results in another variation of the single-phase , single-channel queuing model.

In a finite queue , the length of the queue is limited .

The basic single-server model must be modified to consider the finite queuing system. It should be noted that for this case, the service rate does not have to exceed the arrival rate ( µ > l ) in order to obtain steady-state conditions. The resulting operating characteristics, where M is the maximum number in the system, are as follows :

---

[Page 588]

Because P _n is the probability of n units in the system, if we define M as the maximum number allowed in the system, then P _M (the value of P _n for n = M ) is the probability that a customer will not join the system. The remaining equations are

As an example of the single-server model with finite queue, consider Metro Quick Lube, a one-bay service facility located next to a busy highway in an urban area. The facility has space for only one vehicle in service and three vehicles lined up to wait for service. There is no space for cars to line up on the busy adjacent highway , so if the waiting line is full (three cars), prospective customers must drive on.

The mean time between arrivals for customers seeking lubrication service is 3 minutes. The mean time required to perform the lube operation is 2 minutes. Both the interarrival times and the service times are exponentially distributed. As stated previously, the maximum number of vehicles in the system is four. The operating characteristics are computed as follows:

l = 20

µ = 30

M = 4

First, we will compute the probability that the system is full and the customer must drive on, P _M . However, this first requires the determination of P , as follows:

Next, to compute the average queue length, L _q , the average number of cars in the system, L , must be computed, as follows:

---

[Page 589]

Management Science Application: Providing Telephone Order Service in the Retail Catalog Business

In the TIME OUT box for A. K. Erlang on page 575 it was noted that the origin of queuing analysis was in telephone congestion problems in the early 1900s. Today queuing analysis is still an important tool in analyzing telephone service in catalog phone sales, one of the largest retail businesses in the United States. In a recent year, one of the largest catalog sales companies, L.L.Bean, with over $1 billion in annual sales, received more than 12 million customer calls. During the peak holiday season , its 3,000 customer service representatives took more than 140,000 customer calls on its busiest days. Lands' End, the 15th largest mail-order company in the United States, with sales of over $1.3 billion, handles more than 15 million phone calls each year. On an average day, its 300-plus phone lines handle between 40,000 and 50,000 calls, and during the weeks prior to Christmas, it expands to more than 1,100 phone lines, to handle more than 100,000 phone calls daily. One of the key factors in maintaining a successful catalog phone order system is to provide prompt service; if customers have to wait too long before talking to a customer service representative, they may hang up and not call back. Catalog companies such as L.L.Bean and Lands' End often use queuing analysis to make a number of decisions related to order processing, including the number of telephone trunk lines and customer service representatives they need during various hours of the day and days of the year, the amount of computing capacity they need to handle call volume, the number of workstations and the amount of equipment needed, and the number of full-time and part-time customer service representatives to hire and train.

To compute the average waiting time, W _q , the average time in the system, W , must be computed first:

Computer Solution of the Finite Queue Model with Excel

The Excel spreadsheet solution to the Metro Quick Lube example problem with a finite queue is shown in Exhibit 13.6. Notice that the formula for P in cell D7 is shown on the formula bar at the top of the spreadsheet. The formula for L in cell D9 is shown in the callout box.

Exhibit 13.6.

(This item is displayed on page 590 in the print version)

Note that Excel QM also has a spreadsheet macro for the finite queue model that can be accessed similarly to the single-server model in Exhibit 13.2.

---

[Page 590]

Computer Solution of the Finite Queue Model with QM for Windows

The single-server model with finite queue can be solved with QM for Windows. Exhibit 13.7 shows the model solution screen for our Metro Quick Lube example.

Exhibit 13.7.