# Poisson Probability Distribution Functions and Call Waiting Times Mathematical Background:

The Poisson Probability Cumulative Density function is useful in many applications where the mean of a certain activity is known and questions are arised regarding the probability of certain ranges of values for the stochastic activity.  There are many examples of the applications of this distribution including product warranty ranges, productivity bonus expense estimations, and the probability of  a certain range of items returned for a refund for a manufacturing organization.

One important question that a call center might want to ask is the number of minutes that it keeps its customers waiting for a customer service representative.  This measurement is important to customer satisfaction and cost reduction, both of which are components to the companies overall profitability.  Overestimating the waiting time will result in the company hiring additional employees and incurring an additional cost which might not be worth the reduction in waiting time.  A similar problem arises when the company underestimates the time spent by customers waiting-the savings in wage expenses are not offset by the dissatisfaction of customers having to wait a long time to be heard and consequently increasing the firms cost by initiating a chargeback.  Chargebacks are credits issued to customers by their credit card issuing bank as a result of a claim by the customer as to the validity of charges present on their credit cards.  High chargeback ratios places the contractual agreements between merchants/service providers and major credit cards at risk of being rescinded.

The optimum average number of minutes that a company would want its customers to wait depends on the company cost structure, customer preferences, and the overall business model of the organization.  What the Poisson Probability Distribution model can do is answer sophisticated questions about the probability of waiting time ranges which aids the call center management in estimating the cost and benefits in an accurate way.

The Poisson Probability Distribution Formula where lambda is the 1 divided by the average waiting time for customers and x is the value that you are trying to calculate.  The mathematical proof that lambda is equal to the 1 divided by the average waiting time is given by using the maximum likelihood estimate of the parameter lambda, the proof is:   This formula above is a discrete probaility distribution so you can calculate individual probabilities.  One question that can be answered is, “If the average waiting time is 10 minutes then what is the probability that a random customer has their call answered in 6 minutes?

A clearer picture of the distribution arises then we ask question about the likelihood of waiting, say 6-10 minutes for a customer service representative.  This question involves the cummulative density probability function of the Poisson Distribution.  It is simply the summation of the probability density function where the limits of summation are the ranges of interest. $\sum\frac{{e^{ - \lambda } \lambda ^x }}{{x!}}$

Here is an example:

Suppose that a company had an average call waiting time of 15 minutes and that the owner wants to know the probability of someone waiting on the phone longer than 20 minutes.   He believes that after 20 minutes of waiting the average person will hang up the phone and perform a chargeback through their bank with a 50% probability.  Assuming that the customers who wait more than 20 minutes are the only source of chargebacks for the manager and that the manager has 450 customers per month.  The associated cost per chargeback is $35.00 per initiation. The manager ask you to estimate the number of chargebacks he will be responsible for on on average given this information and their total cost to his organization. The probability of waiting longer than 20 minutes-after some calculations using the Poisson Probability Distribution-is 8.29%. If 8.29% of the total customers will wait longer than 20 minutes then a total of 37.33 customers will wait longer than 20 minutes. From these 37 customers approximately half will initiate a chargeback. The final answer to the call manager would be 19 chargebacks should be expected given the information in the problem. The total cost of these chargebacks is$653.27 month.

The manager is not happy with this number and would like to reduce the expected value of chargebacks and their associated cost.  He concludes that he can hire a 1 more call center operators on a part-time  basis and reduce the average call waiting time (lambda) by 2 minutes to an average waiting time of 13 minutes.  The cost of hiring the new employees is $500.00 per month and you are called to task again to evaluate to feasibility of hiring this new operator based on the reduced number of chargebacks, essentially a cost benefit analysis involving probability. After running a series of calculations using the Poisson Distribution you conclude that the probability of waiting more than 20 minutes is reduced to 2.5% therefore the total number of customers waiting more than 20 minutes is 11.2 of which 5.6 will initiate a chargeback at a cost of$196.98.  The total reduction in expected chargeback initiation expenses is $456.29 and the total increase in wage expenses to hire the part time employee is$500.00 per month.  The recommendation to the manager would be that hiring an additional person would result in an expected loss of \$47.51 so it is not economically feasible to hire another person to reduce the average call waiting time.  The results of this experiement could have gone either way and it is quite possible that it would be economically feasible to hire an additional member to the call center team.

Calculations on Excel:

The numbers generated above were calculated quickly and easily using Microsoft Excel.

To generate the possibility of waiting longer than 20 minutes given an average waiting time of 15 minutes we used the possion(Time under Investigation (20),Average Waiting Time, True) function in Excel. The Excel input would look like = 1 – poisson (20,15, TRUE), then you would get this value and multiply it by the total number of customers to get the expected number of customers waiting longer then 20 minutes.  The Excel input for the total number of customers waiting longer than 20 minutes is based on the average waiting times is = [1 – poisson(20,15,TRUE)]*Number of Sales. The probability of a chargeback is 50%  for customers waiting for more than 20 minutes and the cost of each chargeback is multiplied by this number.  So the final Excel output for the total expected cost of chargebacks per month given the average waiting time of 15 minutes given that half the customers that wait for longer than 20 minutes will chargeback is,

={[1-poisson(20,15,TRUE)]*NumberofSales*((1/2)*CostofChargeback}