An Algebraic Proof of the Relation of Markov Fluid Queues and QBD Processes

Abstract

We study the relation of Markov fluid queues and QBD processes in this paper. Ahn and Ramaswami presented results about this relation and provided a stochastic interpretation based reasoning in [1]. In the current work, first we provide an algebraic proof for that relation.

After that, we present a negative result about the potential extension of the QBD based analysis Markov fluid queues to Markov fluid queues with two buffers. We present a 2-dimensional QBD process, which could be a candidate for describing the stationary behaviour of the related Markov fluid queue, but it turns out that the QBD based behaviour is different from the one of the Markov fluid queue.

A Proof of Theorem 2

Using $\textbf{P}_{++}=\textbf{Q}_{++}/\lambda +\textbf{I}$, $\textbf{P}_{+-}=\textbf{Q}_{+-}/\lambda$, $\textbf{P}_{--}=\textbf{Q}_{--}/\lambda +\textbf{I}$, $\textbf{P}_{-+}=\textbf{Q}_{-+}/\lambda$, for $i \ge j \ge 1$, Equations (45)–(50) can be simplified to

$$\begin{aligned} - p_{00}^- \textbf{Q}_{--} &= \lambda r_2 ( p_{10}^- + p_{11}^-), \end{aligned}$$

(55)

$$\begin{aligned} p_{i0}^- (\lambda r_2\textbf{I}\!-\!\textbf{Q}_{--}) &= \lambda (1\!-\!r_2) p_{i1}^- + \lambda r_2 (p_{i+1,0}^-+p_{i+1,1}^-) , \end{aligned}$$

(56)

$$\begin{aligned} \lambda p_{ij}^+&= p_{i-1,j-1}^+ (\lambda \textbf{I}+ \textbf{Q}_{++}) + p_{i-1,j-1}^- \mathbf {Q_{-+}} , \end{aligned}$$

(57)

$$\begin{aligned} p_{ij}^- (\lambda \textbf{I}\!-\!\textbf{Q}_{--})&= p_{ij}^+ \mathbf {Q_{+-}} + \lambda r_2 p_{i+1,j+1}^- \nonumber \\ & ~~~~+ \mathcal {I}_{\{j<i\}} \lambda (1\!-\!r_2) p_{i,j+1}^-+ \mathcal {I}_{\{j=i\}} p_{i,j+1}^- \mathbf {Q_{+-}} , \end{aligned}$$

(58)

$$\begin{aligned} \lambda p_{1,2}^+&= p_{0,0}^- \mathbf {Q_{-+}}, \end{aligned}$$

(59)

$$\begin{aligned} \lambda p_{i+1,i+2}^+&= p_{i,i+1}^+ (\lambda \textbf{I}+\textbf{Q}_{++} ). \end{aligned}$$

(60)

Multiplying (60) by $\frac{\lambda ^{i} x^{i-1} e^{-\lambda x}}{(i-1)!}$ and summing up from $i=1$ to $\infty$ gives

$$\begin{aligned} \sum _{i=1}^{\infty } \frac{\lambda ^{i} x^{i-1} e^{-\lambda x}}{(i-1)!} \lambda p_{i+1,i+2}^+&= \sum _{i=1}^{\infty } \frac{\lambda ^{i} x^{i-1} e^{-\lambda x}}{(i-1)!} p_{i,i+1}^+ (\lambda \textbf{I}+\textbf{Q}_{++} ), \end{aligned}$$

(61)

$$\begin{aligned} \frac{d}{dx}\tilde{V}(x) + \lambda \tilde{V}(x) &= \tilde{V}(x) (\lambda \textbf{I}+\textbf{Q}_{++} ), \end{aligned}$$

(62)

which results in (51).

By definition $\tilde{W}(0,x)^\pm =\sum _{i=1}^{\infty } \frac{\lambda ^i x^{i-1} e^{-\lambda x}}{(i-1)!} \lambda p_{i,1}^\pm .$ Multiplying (56) by $\frac{\lambda ^i x^{i-1} e^{-\lambda x}}{(i-1)!}$ and summing up from $i=1$ to $\infty$ gives

$$\begin{aligned} &\sum _{i=1}^{\infty } \frac{\lambda ^i x^{i-1} e^{-\lambda x}}{(i-1)!} p_{i0}^- (\lambda r_2\textbf{I}\!-\!\textbf{Q}_{--}) = \sum _{i=1}^{\infty } \frac{\lambda ^i x^{i-1} e^{-\lambda x}}{(i-1)!} \left( \lambda (1\!-\!r_2) p_{i1}^- + \lambda r_2 (p_{i+1,0}^-+p_{i+1,1}^-)\right) , \\ &\tilde{U}(x) (\lambda r_2\textbf{I}\!-\!\textbf{Q}_{--}) = \underbrace{ \sum _{i=1}^{\infty } \frac{\lambda ^i x^{i-1} e^{-\lambda x}}{(i-1)!} \lambda (1\!-\!r_2) p_{i1}^-}_{(1\!-\!r_2)\tilde{W}(0,x)^-} + \underbrace{\sum _{i=1}^{\infty } \frac{\lambda ^i x^{i-1} e^{-\lambda x}}{(i-1)!} \lambda r_2 p_{i+1,1}^-}_{r_2 \left( \frac{d}{dx}\tilde{W}(0,x)^- + \lambda \tilde{W}(0,x)^- \right) } \\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \underbrace{\sum _{i=1}^{\infty } \frac{\lambda ^i x^{i-1} e^{-\lambda x}}{(i-1)!} \lambda r_2 p_{i+1,0}^-}_{r_2 \left( \frac{d}{dx}\tilde{U}(x) + \lambda \tilde{U}(x)\right) }, \end{aligned}$$

which results in (51).

For the computation of $\tilde{W}(x,y)$ we need the following lemma.

Lemma 1

The derivatives of $\tilde{W}(x,y)^\pm =\sum _{i=1}^{\infty } \sum _{j=1}^{i} \frac{\lambda ^i y^{i-1} e^{-\lambda y}}{(i-1)!} \frac{\lambda ^j x^{j-1} e^{-\lambda x}}{(j-1)!} p_{i,j}^\pm$ satisfy

$$\begin{aligned} \frac{\partial }{\partial y} \tilde{W}(x,y)^\pm + \lambda \tilde{W}(x,y)^\pm = \sum _{i=2}^{\infty } \sum _{j=1}^{i} \frac{\lambda ^i y^{i-2} e^{-\lambda y}}{(i-2)!} \frac{\lambda ^j x^{j-1} e^{-\lambda x}}{(j-1)!} p_{i,j}^\pm ~, \end{aligned}$$

(63)

$$\begin{aligned} \frac{\partial }{\partial x} \tilde{W}(x,y)^\pm + \lambda \tilde{W}(x,y)^\pm & = \sum _{i=1}^{\infty } \sum _{j=2}^{i} \frac{\lambda ^i y^{i-1} e^{-\lambda y}}{(i-1)!} \frac{\lambda ^j x^{j-2} e^{-\lambda x}}{(j-2)!} p_{i,j}^\pm \end{aligned}$$

(64)

$$\begin{aligned} & = \sum _{i=2}^{\infty } \sum _{j=2}^{i-1} \frac{\lambda ^{i-1} y^{i-2} e^{-\lambda y}}{(i-2)!} \frac{\lambda ^j x^{j-2} e^{-\lambda x}}{(j-2)!} p_{i-1,j}^\pm ~, \end{aligned}$$

(65)

and

$$\begin{aligned} &\frac{\partial }{\partial x} \frac{\partial }{\partial y} \tilde{W}(x,y)^\pm +\lambda \frac{\partial }{\partial y} \tilde{W}(x,y)^\pm + \lambda \frac{\partial }{\partial x} \tilde{W}(x,y)^\pm + \lambda ^2 \tilde{W}(x,y)^\pm \\ \nonumber &= \sum _{i=2}^{\infty } \sum _{j=2}^{i} \frac{\lambda ^i y^{i-2} e^{-\lambda y}}{(i-2)!} \frac{\lambda ^j x^{j-2} e^{-\lambda x}}{(j-2)!}p_{i,j}^\pm . \end{aligned}$$

(66)

Proof

The statements of the lemma can be obtained by substituting the definition of $\tilde{W}(x,y)^\pm$. We omit the details of the proof here.

Multiplying (57) by $\frac{\lambda ^{i-1} y^{i-2} e^{-\lambda y}}{(i-2)!}\frac{\lambda ^{j-1} x^{j-2} e^{-\lambda x}}{(j-2)!}$ and summing up from $i=2$ to $\infty$ and $j=2$ to i and utilizing (66) gives

$$\begin{aligned} &\underbrace{\sum _{i=2}^{\infty } \sum _{j=2}^{i} \frac{\lambda ^{i-1} y^{i-2} e^{-\lambda y}}{(i-2)!}\frac{\lambda ^{j-1} x^{j-2} e^{-\lambda x}}{(j-2)!} \lambda p_{ij}^+}_{\frac{1}{\lambda } \frac{\partial }{\partial x} \frac{\partial }{\partial y} \tilde{W}(x,y)^+ + \frac{\partial }{\partial y} \tilde{W}(x,y)^+ + \frac{\partial }{\partial x} \tilde{W}(x,y)^+ + \lambda \tilde{W}(x,y)^+} \\ &~~~~~~~~~~~= \underbrace{\sum _{i=2}^{\infty } \sum _{j=2}^{i} \frac{\lambda ^{i-1} y^{i-2} e^{-\lambda y}}{(i-2)!}\frac{\lambda ^{j-1} x^{j-2} e^{-\lambda x}}{(j-2)!} \left( p_{i-1,j-1}^+ (\lambda \textbf{I}+ \textbf{Q}_{++}) + p_{i-1,j-1}^- \mathbf {Q_{-+}}\right) }_{\tilde{W}(x,y)^+ (\lambda \textbf{I}+ \textbf{Q}_{++}) + \tilde{W}(x,y)^- \mathbf {Q_{-+}}} \end{aligned}$$

which results in (53).

For $2\le j \le i$ we rewrite (58) as

$$\begin{aligned} &p_{i-1,j-1}^- (\lambda \textbf{I}\!-\!\textbf{Q}_{--})- p_{i-1,j-1}^+ \mathbf {Q_{+-}} \\ \nonumber &~~~~= \lambda r_2 p_{i,j}^- + \mathcal {I}_{\{j<i\}} \lambda (1\!-\!r_2) p_{i-1,j}^-+ \mathcal {I}_{\{j=i\}} p_{i-1,j}^- \mathbf {Q_{+-}} , \end{aligned}$$

(67)

Multiplying (67) by $\frac{\lambda ^{i-1} y^{i-2} e^{-\lambda y}}{(i-2)!}\frac{\lambda ^{j-1} x^{j-2} e^{-\lambda x}}{(j-2)!}$ and summing up from $i=2$ to $\infty$ and $j=2$ to i and utilizing Lemma 1 gives

$$\begin{aligned} &\underbrace{\sum _{i=2}^{\infty } \sum _{j=2}^{i} \frac{\lambda ^{i-1} y^{i-2} e^{-\lambda y}}{(i-2)!}\frac{\lambda ^{j-1} x^{j-2} e^{-\lambda x}}{(j-2)!} p_{i-1,j-1}^-}_{\tilde{W}(x,y)^-} (\lambda \textbf{I}\!-\!\textbf{Q}_{--}) \\ &~~~- \underbrace{\sum _{i=2}^{\infty } \sum _{j=2}^{i} \frac{\lambda ^{i-1} y^{i-2} e^{-\lambda y}}{(i-2)!}\frac{\lambda ^{j-1} x^{j-2} e^{-\lambda x}}{(j-2)!} p_{i-1,j-1}^+}_{\tilde{W}(x,y)^+} \textbf{Q}_{+-} \\ &~= \underbrace{\sum _{i=2}^{\infty } \sum _{j=2}^{i} \frac{\lambda ^{i-1} y^{i-2} e^{-\lambda y}}{(i-2)!}\frac{\lambda ^{j-1} x^{j-2} e^{-\lambda x}}{(j-2)!} \lambda r_2 p_{i,j}^-}_{r_2\left( \frac{1}{\lambda } \frac{\partial }{\partial x} \frac{\partial }{\partial y} \tilde{W}(x,y)^- + \frac{\partial }{\partial y} \tilde{W}(x,y)^- + \frac{\partial }{\partial x} \tilde{W}(x,y)^- + \lambda \tilde{W}(x,y)^- \right) } \\ &~~~+ \underbrace{\sum _{i=2}^{\infty } \sum _{j=2}^{i-1} \frac{\lambda ^{i-1} y^{i-2} e^{-\lambda y}}{(i-2)!}\frac{\lambda ^{j-1} x^{j-2} e^{-\lambda x}}{(j-2)!} \lambda (1\!-\!r_2) p_{i-1,j}^- }_{(1\!-\!r_2)\left( \frac{\partial }{\partial x} \tilde{W}(x,y)^- + \lambda \tilde{W}(x,y)^-\right) } \\ &~~~+ \underbrace{\sum _{i=2}^{\infty } \frac{\lambda ^{i-1} y^{i-2} e^{-\lambda y}}{(i-2)!}\frac{\lambda ^{i-1} x^{i-2} e^{-\lambda x}}{(i-2)!} p_{i-1,i}^- }_{\varepsilon (x,y)}\mathbf {Q_{+-}}, \end{aligned}$$

that is

$$\begin{aligned} &\tilde{W}(x,y)^- (\lambda \textbf{I}\!-\!\textbf{Q}_{--}) - \tilde{W}(x,y)^+ \textbf{Q}_{+-} \\ & ~~~~~= r_2\left( \frac{1}{\lambda } \frac{\partial }{\partial x} \frac{\partial }{\partial y} \tilde{W}(x,y)^- + \frac{\partial }{\partial y} \tilde{W}(x,y)^- + \frac{\partial }{\partial x} \tilde{W}(x,y)^- + \lambda \tilde{W}(x,y)^- \right) \\ &~~~~+ (1\!-\!r_2)\left( \frac{\partial }{\partial x} \tilde{W}(x,y)^- + \lambda \tilde{W}(x,y)^-\right) + \varepsilon (x,y)\mathbf {Q_{+-}}, \end{aligned}$$

which results in (54).