Students arrive for the much dreaded JEE Advanced according to a Poisson process with rate lambda. For sanitization process they must stand in a queue,each student can take different time or same time compared with some other student for sanitization. Let us denote the time taken by i-th student as $X}_{i$ .

$X}_{i$ are independent identically distributed random variables. We assume that $X}_{i$ takes integer values in range 1,2, ... , n, with probabilities $p}_{1},{p}_{2},\dots ,{p}_{n$ . Find the PMF for $N}_{t$ , the number of students in sanitization queue at time t.