I am trying to formulate one equation.
Let $Q(t)$ represent the contents of a single-server discrete time queueing system defined over integer time slots $t \in \{0, 1, 2, . . .\}$. Specifically, the initial state $Q(0)$ is assumed to be a non-negative real valued random variable. Future states are driven by stochastic arrival and server processes $a(t)$ and $b(t)$ according to the following dynamic equation:
$$Q(t+1)=\max[Q(t)−b(t),0]+a(t)\quad \text{for}\quad t\in\{0,1,2,...\}.$$
The value of $a(t)$ represents the amount of new work that arrives on slot $t$, $b(t)$ represents the amount of work the server of the queue can process on slot $t$.
Now if I want to say, for each arrival of $a(t)$ I want to put $+1$, and for each time processing the work by server, I want to give $-1$ to $b(t)$, so that I want to see after $\{t+1, t+2,.......,t+K\}$, this queuing systems posses positive or negative value. Then how should I formulate the equation?
