Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplines

Abstract

This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths of different customer classes under DPS undergoes a state-space collapse when subject to heavy-traffic scaling. That is, the limiting distribution of the queue-length vector is shown to be exponential, times a deterministic vector. The distribution of the scaled workload, as derived for general service disciplines, is a key quantity in the proof of the state-space collapse.

Appendix: Proof of Theorem 3.3

The proof Theorem 3.3 is based on Theorem 4 in [12], which can be adapted to our model as follows:

We start with notation and some preliminaries. Let $\Lambda = \text {diag}(\lambda _1,\ldots ,\lambda _D)$, $\bar{H}(s) = \text {diag}(1 - h_1(s),\ldots ,1 - h_D(s))$, $C= \text {diag}(c_1,\ldots ,c_D)$ and $\varvec{p}_0 = (p_{0,1},\ldots ,p_{0,D})$. Furthermore $\bar{H}_1$ and $\bar{H}_2$ are the diagonal matrices corresponding to the moments $h_{d1}$ and $h_{d2}$, respectively, for $d=1,\ldots ,D$. Recall Eq. (3), $[Q\cdot \varvec{a}]_d = c_d - \lambda _d h_{d1} -c_\infty (1-\rho _\infty )$. We will now construct a partial inverse of Q to make it easier to find a vector $\varvec{a}$ which solves this equation. Let $Q_1$ and R be matrices such that

$$\begin{aligned} Q_1= \begin{pmatrix} q_{22} &{} q_{23} &{} \cdots &{} q_{2D} \\ \vdots &{} &{} &{}\\ q_{D2} &{} q_{D3} &{} \cdots &{} q_{DD} \end{pmatrix}, \quad R= \begin{pmatrix} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{}&{}&{}\\ \vdots &{} Q_1^{-1} &{} \\ 0 &{} &{}&{} \end{pmatrix}. \end{aligned}$$

Then $\det Q_1 \ne 0$ and, due to Q being a generator (for more details, see [12]), we have

$$\begin{aligned} QR = \begin{pmatrix} 0 &{} \frac{-\pi _2}{\pi _1} &{} \frac{-\pi _3}{\pi _1} &{} \cdots &{} \frac{-\pi _D}{\pi _1} \\ 0 &{} 1 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} \cdots \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \ddots \\ 0 &{} 0 &{} \cdots &{} \cdots &{} 1 \end{pmatrix}. \end{aligned}$$

It follows that for any vector $\varvec{x}$, it holds that

$$\begin{aligned} \varvec{x}QR = \varvec{x} - \frac{x_1}{\pi _1} \varvec{\pi }. \end{aligned}$$

(33)

Then it can be verified with straightforward calculations that

$$\begin{aligned} \varvec{a}&= (a_1,\ldots ,a_D) = R(c_1 - \lambda _1 h_{11} - c_\infty (1-\rho _\infty ),\ldots ,c_D - \lambda _D h_{D1} - c_\infty (1-\rho _\infty ))^T \nonumber \\&= R[C - \Lambda \bar{H}_1]\varvec{e} - c_\infty (1-\rho _\infty )R\varvec{e} \nonumber \\&= R[C - \Lambda \bar{H}_1]\varvec{e} - \varvec{r} \end{aligned}$$

(34)

is a possible solution vector, with $\varvec{r} := c_\infty (1-\rho _\infty )R\varvec{e}$.

Define the vector $\varvec{\varphi } = (\varphi _1(s),\ldots ,\varphi _D(s))$ and write Eq. (5) in matrix-vector terms:

$$\begin{aligned} \varvec{\varphi }(s) Q = \varvec{\varphi }(s)[\Lambda \bar{H}(s) - sC] + s \varvec{p}_0 C. \end{aligned}$$

(35)

Observe that, according to Eq. (7),

$$\begin{aligned} \varvec{p}_0 C\varvec{e} = c_\infty (1-\rho _\infty ). \end{aligned}$$

Now multiply from the right both sides of the new vector-matrix equation, Eq. (35), with a D-dimensional vector of 1’s, $\varvec{e}$, to obtain

$$\begin{aligned} \varvec{\varphi }(s)[\Lambda \bar{H}(s) - sC]\varvec{e} + s c_\infty (1-\rho _\infty ) = 0. \end{aligned}$$

(36)

Multiply from the right both sides of Eq. (35) with the matrix R to get

$$\begin{aligned} \varvec{\varphi }(s)QR = \varvec{\varphi }(s)[\Lambda \bar{H}(s) - sC]R + s \varvec{p}_0 C R. \end{aligned}$$

Rewrite this equation by using the property of Eq. (33) to obtain

$$\begin{aligned} \varvec{\varphi }(s) = \frac{\varphi _1(s)}{\pi _1} \varvec{\pi } + \varvec{\varphi }(s)[\Lambda \bar{H}(s) - sC]R + s \varvec{p}_0 C R. \end{aligned}$$

(37)

Iterate Eq. (37) with itself by inserting $\varvec{\varphi }(s)$ into the right-hand side of the equation to obtain, after some algebraic transformations,

$$\begin{aligned} \varvec{\varphi }(s) = \frac{\varphi _1(s)}{\pi _1} \varvec{\pi }[I + G(s)R] + \varvec{y}(s), \end{aligned}$$

(38)

with $G(s) := \Lambda \bar{H}(s) - sC$ and

$$\begin{aligned} \varvec{y}(s) := \varvec{\varphi }(s)[G(s)R]^2 + s\varvec{p}_0 CR[G(s)R + I]. \end{aligned}$$

(39)

Substitute Eq. (38) into Eq. (36) to obtain an expression for $\varphi _1(s)$:

$$\begin{aligned} 0&= \left[ \frac{\varphi _1(s)}{\pi _1} \varvec{\pi }[I + G(s)R] + \varvec{y}(s)\right] \cdot G(s)\varvec{e} + s c_\infty (1-\rho _\infty ) \nonumber \\&= \frac{\varphi _1(s)}{\pi _1}\left[ \varvec{\pi } G(s)\varvec{e} + \varvec{\pi }G(s)RG(s)\varvec{e} \right] + \varvec{y}(s)G(s)\varvec{e} + s c_\infty (1-\rho _\infty ) \nonumber \\&= \frac{\varphi _1(s)}{\pi _1}\left[ B_2(s) + B_3(s)\right] + B_1(s), \end{aligned}$$

(40)

with $B_1(s) = \varvec{y}(s)G(s)\varvec{e} + s c_\infty (1-\rho _\infty )$, $B_2(s) = \varvec{\pi }G(s)\varvec{e}$ and $B_3(s) = \varvec{\pi }G(s)RG(s)\varvec{e}$.

The next step is to insert the scaling $s \mapsto s/N$ for each term. Recall that using the heavy-traffic parametrization introduced in Sect. 2, we have $(1-\rho _\infty )=1/N$. Now observe that, as $N\rightarrow \infty$,

$$\begin{aligned} \frac{\bar{H}(s/N)}{s/N} \rightarrow \bar{H}_1, \quad \frac{\bar{H}_1s/N - \bar{H}\left( s/N\right) }{(s/N)^2} \rightarrow \frac{\bar{H}_2}{2}. \end{aligned}$$

Therefore the limit

$$\begin{aligned} \frac{G(s/N)}{s/N} \rightarrow \Lambda \bar{H}_1 - C, \end{aligned}$$

is a constant, and since $|\varvec{\varphi }(s/N)| \le 1$ and $\varvec{p}^{(N)}_0 \rightarrow 0$ (see Eq. (7)), we have

$$\begin{aligned} \frac{\varvec{y}(s/N)}{s/N}&= \varvec{\varphi }(s/N)\frac{[G(s/N)R]^2}{s/N} + \varvec{p}^{(N)}_0 CR[G(s/N)R + I] \\&= \varvec{\varphi }(s/N)\left( \frac{s}{N}\right) \left[ \frac{G(s/N)}{s/N}R\right] ^2 + \varvec{p}^{(N)}_0 CR[G(s/N)R + I] \\&\rightarrow 0, \end{aligned}$$

as $N\rightarrow \infty$. Combining the above we obtain

$$\begin{aligned} B_1(s/N)&= \varvec{y}(s/N)G(s/N)\varvec{e} + sc_\infty (1-\rho _\infty )/N \\&= sc_\infty (1 - \rho _\infty )/N + o(N^{-2}) = sc_\infty /N^2 + o(N^{-2}). \end{aligned}$$

Then

$$\begin{aligned} B_2(s/N)&= \varvec{\pi } \left[ \Lambda \bar{H}\left( s/N\right) - sC/N \right] \varvec{e} \\&= \varvec{\pi } \Lambda \left[ \bar{H}\left( s/N\right) - \bar{H}_1s/N\right] \varvec{e} + \varvec{\pi }\left[ \Lambda \bar{H}_1s/N - sC/N \right] \varvec{e} \\&= \varvec{\pi } \Lambda \frac{\bar{H}\left( s/N\right) - \bar{H}_1s/N}{(s/N)^2}(s/N)^2\varvec{e} - sc_\infty (1 - \rho _\infty )/N\\&= -\varvec{\pi } \Lambda \frac{\bar{H}_2}{2}(s/N)^2\varvec{e} + o(N^{-2}) - sc_\infty /N^2\\&= -(s/N)^2\sum _{d=1}^D \pi _d\lambda _d h_{d2}/2 - sc_\infty /N^2 + o(N^{-2}), \end{aligned}$$

since $\varvec{\pi } \Lambda \bar{H}_1 \varvec{e} = \sum _{d=1}^D \pi _d\lambda _d h_{d1} = \rho _\infty c_\infty$. Furthermore,

$$\begin{aligned} B_3(s/N)&= \varvec{\pi } \left[ \Lambda \bar{H}\left( s/N\right) - C s/N \right] R \left[ \Lambda \bar{H}\left( s/N\right) - Cs/N \right] \varvec{e} \\&= \varvec{\pi } (s/N)^2\left[ \Lambda \frac{\bar{H}\left( s/N\right) }{(s/N)} - C\right] R \cdot \left[ \Lambda \frac{\bar{H}\left( s/N\right) }{(s/N)} - C \right] \varvec{e} \\&= \varvec{\pi } (s/N)^2 \left[ \Lambda \bar{H}_1 - C \right] R \left[ \Lambda \bar{H}_1 - C\right] \varvec{e} + o(N^{-2}) \\&= -\varvec{\pi } (s/N)^2 \left[ \Lambda \bar{H}_1 - C \right] (\varvec{a} + \varvec{r}) + o(N^{-2}) \\&= -\sum _d \pi _d\left[ (a_d+r_d) (\lambda _d h_{d1} - c_d)\right] (s/N)^2 + o(N^{-2}), \end{aligned}$$

due to $R \left[ C - \Lambda \bar{H}_1\right] \varvec{e} = \varvec{a} + \varvec{r}$; see Eq. (34). Under the heavy-traffic scaling,

$$\begin{aligned} \varvec{r} = c_\infty (1-\rho _\infty )R\varvec{e} = c_\infty N^{-1}R\varvec{e} \end{aligned}$$

is an o(1) term. Observe that

$$\begin{aligned}&-(B_2(s/N) + B_3(s/N)) \\&\quad = (s/N)^2 \sum _{d=1}^D \pi _d[\lambda _d h_{d2}/2 + (a_d + o(1))(\lambda _dh_{d1} - c_d)] + sc_\infty /N^2 + o(N^{-2}). \end{aligned}$$

Rearranging Eq. (40) yields

$$\begin{aligned} \varphi _1(s/N)&= \pi _1\frac{B_1(s/N)}{-(B_2(s/N) + B_3(s/N))} \\&= \pi _1 \frac{c_\infty s/N^2 + o(N^{-2})}{(s/N)^2 \sum _{d=1}^D \pi _d[\lambda _d h_{d2}/2 + a_d(\lambda _dh_{d1} - c_d)] + c_\infty s/N^2 + o(N^{-2}) } \\&= \pi _1 \frac{1 + o(1)}{1 + c_\infty ^{-1} \sum _d \pi _d \left[ \lambda _d h_{d2}/2 + a_d (\lambda _d h_{d1} - c_d)\right] s + o(1)}. \end{aligned}$$

Let M be the desired mean stated in Theorem 3.3, that is,

$$\begin{aligned} M := c_\infty ^{-1}\sum _d \pi _d\left[ \hat{\lambda }_d h_{d2}/2 + a_d (\hat{\lambda }_d h_{d1} - c_d)\right] . \end{aligned}$$

Then, taking the heavy-traffic limit,

$$\begin{aligned} \lim _{N\rightarrow \infty } \varvec{\varphi }(s/N) = \lim _{N\rightarrow \infty } \frac{\varphi _1(s/N)}{\pi _1}\varvec{\pi } = \frac{\varvec{\pi }}{1 + Ms}, \end{aligned}$$

i.e. the LST $\varvec{\varphi }(s)$ converges in distribution to the LST of an exponentially distributed random variable with mean M.