Let us assume that a marketing firm or a company is trying to acquire a new group of customers (in a particular age group or in a particular income group, etc.). In case the company is successful in doing so it makes a profit (say for example through increase in its market share or increase in number of products sold, etc.,) of say  at the present time instance. This  in marketing terms would be called the net contribution at the present time, the company makes to its coffers through its ability by successfully acquiring a particular customer. We also assume that this value of  would be gained by the company in the first or subsequent instance after the customer is acquired/retained. Also consider that there may be many instances or time periods when the customer, does not make any purchase to add value to the company, but that does not stop the company from spending a net amount denoted by , which is at present value and denotes how much money the company/firm is willing to spend in order to retain it customer say through television advertisement, direct marketing, etc. Furthermore, assume that the company believes the probability that the customer will purchase at the end of any particular period is a function of customer's recency only. This denotes the actually number of periods elapsed since the customer's last purchase, i.e., if the customer purchased at the end of last period, then his/her recency value is 1 for the current period. Let us also assume that the probability the customer will purchase at the end of any period is , where  is customer's recency. Thus the probability transition matrix  may be written as given below :

Table 11.2: Transition probability matrix for the brand retention example