Waiting-Time Computational Methods

A manager usually requires that eighty (80%) of the calls are answered within twenty (20) seconds [8]. Thus, the probability that customers wait less than twenty (20) seconds should be at least eighty (80) percent. Mathematically, this is represented, in general, as P(W ≤ t) ≥ α, where t could be twenty seconds, andα could be eighty percent. This quantity is the waiting time distribution and is used to assess the customer service levels received in a call center. Sometimes, this probability is written in terms of the“tail” distribution asP(W ≥t)≤1−α. In other words, the probability that customers wait more than 20 seconds should be less than 20 percent.

There exists a lot of research on customer waiting-times for different types of queueing systems. Although early researchers focused on computing the mean waiting-time, recent ones have derived methods for approximating the waiting- time distribution. For the M/G/1, FCFS queue, the Laplace transform of the waiting-time distribution is determined from the Pollacek-Khinchin (P − K) transform equation for the number of customers in the system. This equation was first published by Khinchin [38] in 1932 and studied by Pollaczek [59] in 1930. Kleinrock [41] and Wolff [73], among others, provide derivations of the

waiting-time distribution Laplace transform using the P −K equations.

Customer waiting-times have been analyzed for theM/G/n, FCFS queue as well. Kingman [40] derives bounds on the mean waiting-time. Newell [55] and Halachmi and Franta [26] use diffusion approaches to compute the mean waiting- time. Unlike the previous researchers, Hokstad [31] computes approximate results for the distribution of the number of customers in the system and the distribution of the waiting-time. Whitt [71] uses the Laplace transform of the waiting-time distribution to predict queueing delays of customers before they enter service. In this case, delay information based on the distribution is seen as more insight- ful than delay information based on the mean. Finally, Kleinrock also derives waiting-time distribution transform for theG/M/n, FCFS queue. He shows that the distribution has an asymptotic exponential form. Although his transform is computed for a conditional waiting-time distribution, given that a customer must queue, the unconditional waiting-time distribution can also be computed.

The waiting-time distribution has also been studied under the priority queue discipline. Jaiswal [33] derives the results for the Laplace transform of the waiting- time distribution for the M/G/n, non-preemptive priority queue. He first com- puted the Laplace transform for the busy period containing customers of higher priority than the given customer. Then, the waiting-time distribution transform can be written as a function of the busy period transform. Kleinrock, Wolff, and Takagi [67] also describe similar results to Jaiswal’s for the M/G/n, non- preemptive priority queue. Takagi inverts the waiting-time distribution for the

M/G/1 queue. However, his resulting formula involves evaluating an infinite se- ries. Thus, obtaining useful values for the waiting-time distribution in queueing applications can be rather complex.

One of the earliest results for the waiting-time distribution for a priority queue was published by Cobham [13] and Kesten and Runnenburg [37]. Shortly after, Takacs [66] expanded on their techniques and provided a simple method to determine the waiting-time distribution for a customer in any class p. For the stationary M/G/1, preemptive and non-preemptive priority, queues, he com- puted the waiting-distribution as a function of the moments of the service time distribution. Another early result was published by Davis [16]. He gives an ex- plicit formula for the waiting-time distribution for an arbitrary customer in the

M/M/n, non-preemptive priority queue. He not only derived the Laplace trans-

form, but also inverted this transform using contour integration. However, his subsequent waiting-time distribution formula was somewhat complex, like Tak- agi’s. Thus, computing values of the waiting-time distribution is as easy as other methods, when applied to real-world problems, such as call centers. Williams [72] expands on Hokstad’s earlier work on theM/G/n, FCFS queue. He computes the waiting-time distribution transform for high and low priority customers in a two- customer, non-preemptive priority setting. Also, for theM/M/n, non-preemptive priority queue, Kella and Yechiali [36] derives a waiting time distribution trans- form result similar to Davis’s. They use a less elaborate method that uses the probabilistic equivalence of the waiting times in the M/G/1 queue with server vacations to those in their queue setting.

Finally, there are other types of priority queues that have been studied. Wolff derives the waiting-time distribution transform for high and low priority cus- tomers in the M/G/1, preemptive-resume priority queue. He first computed an ordinary and exceptional-first-service busy period duration transform for his work. Also, Kleinrock [42] derives only the mean waiting-time for M/G/1, dy-

namic priority queue. He uses a dynamic priority service discipline that uses a time-dependent priority structure where priorities increase or decrease linearly over time. Finally, Jackson [32] gives further results for other dynamic priority queues. His results were some of the earliest on dynamic priority queues.