Analysis of a queue with general service demands and correlated service capacities

Abstract

We present the study of a non-classical discrete-time queueing model in which the customers each request a variable amount of service, called their “service demand”, from a server which is able to execute a variable amount of work, called its “service capacity”, during each time slot. We assume that the numbers of arrivals in consecutive time slots and the service demands of consecutive customers form two independent and identically distributed sequences. However, we allow the service capacities in consecutive time slots to be correlated according to a discrete-batch Markovian process. We study this model analytically and obtain expressions for the probability generating function of the steady-state system content and customer delay, as well as their moments and an approximation for their tail probabilities. The results are illustrated with several numerical examples.

Appendix A: Vacations as a special case of the model

As mentioned in our second numerical example in Sect. 8, vacations with phase-type length are simply a special case of the queuing model, and therefore do not need to be modeled separately. Indeed, consider a server for which the service capacities (without vacations) are described by the matrix generating function ${\hat{\mathbf {R}}}(z)$ but which can at the end of every (non-vacation) slot start a vacation with probability p with a duration that follows the discrete phase-type $PH_d(\varvec{\tau },\mathbf {T})$ distribution, i.e., the lengths of the vacations follow the same distribution as the number of steps it takes for the Markov chain with transition (block) matrix

$$\begin{aligned} \begin{bmatrix} \mathbf {T}&\quad (\mathbf {I} - \mathbf {T})\mathbf {1} \\ \mathbf {0}&\quad 1 \end{bmatrix}, \end{aligned}$$

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to go from an initial state sampled from a distribution described by the (row) vector $\varvec{\tau }$ to the absorbing state. During the vacations, the available service capacity is 0.

There are several possibilities for what happens to the state of the service capacity’s background Markov chain during vacations. If this background Markov chain simply keeps advancing as normal, then the system with vacations is a special case of our model where the matrix generating function of the service capacity is given by the block matrix

$$\begin{aligned} \mathbf {R}(z) = \begin{bmatrix} \mathbf {T} \otimes {\hat{\mathbf {R}}}(1)&\quad (\mathbf {I} - \mathbf {T})\mathbf {1} \otimes {\hat{\mathbf {R}}}(1) \\ p \varvec{\tau } \otimes {\hat{\mathbf {R}}}(z)&\quad (1-p) {\hat{\mathbf {R}}}(z) \end{bmatrix}, \end{aligned}$$

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where $\otimes$ denotes the Kronecker product. Indeed, if the background Markov chain is in state $am+b$, with $0 \le b \le m-1$, then that corresponds to the background Markov chain of the server without vacations being in state b, while a corresponds to the phase of the vacation that the server is currently in (if any).

A second possibility is if the state of the background Markov chain of the server is paused during vacations. In that case only a small modification of the expression for $\mathbf {R}(z)$ is required, and we get

$$\begin{aligned} \mathbf {R}(z) = \begin{bmatrix} \mathbf {T} \otimes \mathbf {I}&\quad (\mathbf {I} - \mathbf {T})\mathbf {1} \otimes \mathbf {I} \\ p \varvec{\tau } \otimes {\hat{\mathbf {R}}}(z)&\quad (1-p) {\hat{\mathbf {R}}}(z) \end{bmatrix}. \end{aligned}$$

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A third possibility is if the state of the background Markov chain after a vacation is independent of the state before the vacation, and is instead simply equal to n with probability $[\kappa ]_n$. Then $\mathbf {R}(z)$ is given by

$$\begin{aligned} \mathbf {R}(z) = \begin{bmatrix} \mathbf {T}&\quad (\mathbf {I} - \mathbf {T})\mathbf {1} \otimes \varvec{\kappa } \\ p \varvec{\tau }&\quad (1-p) {\hat{\mathbf {R}}}(z) \end{bmatrix}. \end{aligned}$$

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