## Delivery Time Models

The reply of a respondent is delivered *i* days after the launch of a direct marketing campaign due to the delay rate *q* and the delivery time *d*. Thus, in the geometric response model, the probability *P*_{i} that a respondent’s reply will be received on day *i* is

#### Equation (1-12)

where *P*[*d*_{j}] is the probability mass function of the delivery time. In a special case in which the delivery is instant, the probability distribution becomes

#### Equation (1-13)

In such a case, we can simply have

#### Equation (1-14)

Let us consider three different probability mass functions of *d* with a single parameter. First, suppose that the delivery time *d* has a discrete *uniform* distribution as in Basu, Basu, and Batra (1995);

#### Equation (1-15)

where *u* is the upper limit of the uniform random variable. The delivery is instant if *u*=0. The expected value of the uniform delivery time is

#### Equation (1-16)

It follows from (1-12) and (1-15) that

#### Equation (1-17)

which can be simplified further as

#### Equation (1-18)

Second, suppose that the delivery time *d* has a *geometric* distribution:

#### Equation (1-19)

where *r* is a parameter, 0<*r*<1, to be estimated empirically. If *r* is close to zero, then the delivery time is negligible. The expected value of the geometric random variable is

#### Equation (1-20)

It follows from (1-12) and (1-19) that

#### Equation (1-21)

Third, suppose that the delivery time *d* has a *Poisson* distribution:

#### Equation (1-22)

where *s* is a parameter, *s*>0, to be estimated empirically. The delivery time is negligible if *s* is close to 0. The average delivery time in (1-22) is

#### Equation (1-23)

With the Poisson delivery time, it follows from (1-12) and (1-22) that

#### Equation (1-24)

Figure 1.3 illustrates the three probability distributions in which the average delivery time is *E*[*d*] = 2 days. Among the three distributions, the Poisson delivery in Figure 1.3(c) appears to be the most realistic in most practical situations.

**Figure 1.3** Various delivery time distributions with *E*[*d*]=2 days.

Note that we may consider other discrete probability distributions with more than one parameter. For example, the negative binomial distribution has been widely used in various consumer behavior models (Wagner and Taudes 1987) and in product inspection models (Chun and Sumichrast 2007). However, we restrict our attention to the single-parameter delivery time to have a parsimonious response model. Thus, our geometric response model has only three parameters: response rate, delay rate, and delivery time. All of the parameters have meaningful interpretations. In the next section, we compare the performance of the three delivery time models using weekly response data and propose the best one.