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An analysis of nonstationary coupled queues

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Abstract

We consider a two dimensional time varying tandem queue with coupled processors. We assume that jobs arrive to the first station as a non-homogeneous Poisson process. When each queue is non-empty, jobs are processed separately like an ordinary tandem queue. However, if one of the processors is empty, then the total service capacity is given to the other processor. This problem has been analyzed in the constant rate case by leveraging Riemann Hilbert theory and two dimensional generating functions. Since we are considering time varying arrival rates, generating functions cannot be used as easily. Thus, we choose to exploit the functional Kolmogorov forward equations (FKFE) for the two dimensional queueing process. In order to leverage the FKFE, it is necessary to approximate the queueing distribution in order to compute the relevant expectations and covariance terms. To this end, we expand our two dimensional Markovian queueing process in terms of a two dimensional polynomial chaos expansion using the Hermite polynomials as basis elements. Truncating the polynomial chaos expansion at a finite order induces an approximate distribution that is close to the original stochastic process. Using this truncated expansion as a surrogate distribution, we can accurately estimate probabilistic quantities of the two dimensional queueing process such as the mean, variance, and probability that each queue is empty.

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Acknowledgments

This work is partially supported by a Ford Foundation Fellowship and Cornell University.

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Authors and Affiliations

  1. School of Operations Research and Information Engineering, Cornell University, Ithaca, USA

    Jamol Pender

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  1. Jamol Pender
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Appendix

Appendix

1.1 Hermite polynomials

Lemma 5.1

(Stein [26]). The random variable \(X\) is Gaussian\((0,1)\) if and only if

$$\begin{aligned} E\left[ X \cdot f(X)\right] = E\left[ \frac{d\ }{dX}f(X)\right] , \end{aligned}$$
(5.1)

for all generalized functions \(f\). Moreover, we also have that

$$\begin{aligned} E\left[ h_n(X) \cdot f(X)\right] = E\left[ \frac{d^n\ }{dX^n}f(X)\right] , \end{aligned}$$
(5.2)

where \(h_n(X)\) is the \(n^{th}\) Hermite polynomial.

Let \(X\) and \(Y\) be two i.i.d Gaussian(0,1) random variables (Fig. 8).

Fig. 8
figure 8

Simulated mean and variance (Left). Simulated Skewness and Kurtosis (Right)

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Proposition 5.2

Any \(L^2\) function can be written as an infinite sum of Hermite polynomials of \(X\), i.e.

and

$$\begin{aligned} \mathrm {Cov}[f(X), g(X,Y)]= & {} \sum ^{\infty }_{m=1} \frac{1}{m!} E\left[ \frac{\partial ^{m}f}{\partial ^m X }(X) \right] \\&\cdot ~E\left[ \frac{\partial ^{m}g}{\partial ^m X }(X,Y) \right] \\ \end{aligned}$$

1.2 Calculation of expectation and covariance terms

We define \(\varphi \) and \(\Phi \) to be the density and the cumulative distribution functions, for \(X\) respectively, where

$$\begin{aligned} \varphi (x)\equiv & {} \frac{1}{\root \of {2\pi }} e^{-x^2/2}, \ \ \ \Phi (x) \equiv \int _{-\infty }^x \varphi (y)\, dy,\ \ \text{ and } \nonumber \\ \overline{\Phi }(x)\equiv & {} 1-\Phi (x)= \int _{x}^{\infty } \varphi (y)\ dy. \end{aligned}$$
(5.3)

We begin with some of the simpler expectation terms that only involve the evaluation of the Gaussian tail cdf.

$$\begin{aligned} E[\{ X > \chi _1 \}]= & {} {\mathbb {P}}\left( X > \chi _1 \right) = \overline{\Phi }\left( \chi _1 \right) \\ E[\{ X \cdot \cos \theta + Y \cdot \sin \theta > \chi _2 \}]= & {} {\mathbb {P}}\left( Z > \chi _2 \right) = \overline{\Phi }\left( \chi _2 \right) \end{aligned}$$

Now we use the previous proposition to derive the following expectations. Using the \(L^2\) expansion of the function, we get an infinite series representation for the first line. To move from the second to the third line, we use the fact that the function \(\{ Q_1 > 0 \}\) does not depend on the function \(Y\). Lastly, we use the Hermite polynomial generalization of Stein’s lemma (Fig. 9).

$$\begin{aligned}&\mathrm {E}\left[ \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \sum ^{\infty }_{m=0} \sum ^{\infty }_{n=0} \frac{1}{m!n!} E\left[ \frac{\partial ^{n+m}}{\partial ^m X \partial ^n Y} \{ Q_1 > 0 \} \right] \\&\qquad \cdot ~E\left[ \frac{\partial ^{n+m}}{\partial ^m X \partial ^n Y} \{ Q_2 > 0 \} \right] \\&\quad = \sum ^{\infty }_{m=0} \frac{1}{m!} E\left[ \frac{\partial ^{m}}{\partial ^m X } \{ Q_1 > 0 \} \right] \cdot E\left[ \frac{\partial ^{m}}{\partial ^m X } \{ Q_2 > 0 \} \right] \\&\qquad \mathrm {(since \ Q_1 \ does \ not \ depend \ on \ Y)}\\&\quad = \overline{\Phi }( \chi _1) \cdot \overline{\Phi }( \chi _2) + \phi (\chi _1) \cdot \phi (\chi _2) \cdot \sum ^{\infty }_{m=1} \frac{1}{m!} \\&\qquad \cdot ~h_{m-1}(\chi _1) \cdot h_{m-1}(\chi _2) \cdot \cos ^{m} \theta . \end{aligned}$$
Fig. 9
figure 9

Simulated variance vs. GVA variance of queue 1 (Left). Simulated variance vs. GVA variance of queue 2 (Right)

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The following two expectations can be calculated easily using the previous calculations.

$$\begin{aligned}&\mathrm {E}\left[ \{ Q_1 > 0 \} \cdot \{ Q_2 \le 0 \} \right] \\&\quad = \mathrm {E}\left[ \{ Q_1 > 0 \} \right] - \mathrm {E}\left[ \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad =\overline{\Phi }( \chi _1) - \overline{\Phi }( \chi _1) \cdot \overline{\Phi }( \chi _2) - \phi (\chi _1) \cdot \phi (\chi _2) \\&\qquad \cdot \sum ^{\infty }_{m=1} \frac{1}{m!} \cdot h_{m-1}(\chi _1) \cdot h_{m-1}(\chi _2) \cdot \cos ^{m} \theta \\&\mathrm {E}\left[ \{ Q_1 \le 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \mathrm {E}\left[ \{ Q_2 > 0 \} \right] - \mathrm {E}\left[ \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad =\overline{\Phi }( \chi _2) - \overline{\Phi }( \chi _1) \cdot \overline{\Phi }( \chi _2) - \phi (\chi _1) \cdot \phi (\chi _2) \\&\qquad \cdot \sum ^{\infty }_{m=1} \frac{1}{m!} \cdot h_{m-1}(\chi _1) \cdot h_{m-1}(\chi _2) \cdot \cos ^{m} \theta \end{aligned}$$

Now we begin the calculation of the covariance terms with respect to the first queue length. From the first line to the second we use the property that covariances are invariant to constants. Then, we use the Hermite polynomial expansion property and the Hermite polynomial generalization of Stein’s lemma once again (Fig. 10).

$$\begin{aligned}&\mathrm {Cov}\left[ Q_1, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \mathrm {Cov}\left[ q_1 + \sqrt{v_1} \cdot X, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_1} \cdot \mathrm {Cov}\left[ X, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_1} \cdot \mathrm {E}\left[ X \cdot \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_1} \cdot \sum ^{\infty }_{m=0} \sum ^{\infty }_{n=0} \frac{1}{m!n!} E\left[ \frac{\partial ^{n+m}}{\partial ^m X \partial ^n Y} X \cdot \{ Q_1 > 0 \} \right] \\&\qquad \cdot ~E\left[ \frac{\partial ^{n+m}}{\partial ^m X \partial ^n Y} \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_1} \cdot \sum ^{\infty }_{m=0} \frac{1}{m!} E\left[ \frac{\partial ^{m}}{\partial ^m X } X \cdot \{ Q_1 > 0 \} \right] \\&\qquad \cdot ~E\left[ \frac{\partial ^{m}}{\partial ^m X } \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_1} \cdot \sum ^{\infty }_{m=0} \frac{1}{m!} E\left[ \frac{\partial ^{m}}{\partial ^m X } X \cdot \{ X > \chi _1 \} \right] \\&\qquad \cdot ~E\left[ \frac{\partial ^{m}}{\partial ^m X } \{ X \cdot \cos \theta + Y \cdot \sin \theta > \chi _2 \} \right] \\&\quad = \sqrt{v_1} \cdot \sum ^{\infty }_{m=0} \frac{1}{m!} E\left[ h_m(X) \cdot X \cdot \{ X > \chi _1 \} \right] \\&\qquad \cdot ~E\left[ \frac{\partial ^{m}}{\partial ^m X } \{ X \cdot \cos \theta + Y \cdot \sin \theta > \chi _2 \} \right] \\&\quad = \sqrt{v_1} \cdot \sum ^{\infty }_{m=0} \frac{1}{m!} E\left[ (h_{m+1}(X) + m \cdot h_{m-1}(X) ) \right. \\&\qquad \left. \cdot ~\{ X > \chi _1 \} \right] \cdot E\left[ \frac{\partial ^{m}}{\partial ^m X } \{ X \cdot \cos \theta + Y \cdot \sin \theta > \chi _2 \} \right] \end{aligned}$$
$$\begin{aligned}&\quad = \sqrt{v_1} \cdot \varphi (\chi _1) \cdot \overline{\Phi }(\chi _2) + \sqrt{v_1} \cdot \left( \overline{\Phi }(\chi _1) \right. \\&\qquad \left. +~\chi _1 \cdot \varphi (\chi _1) \right) \cdot \varphi (\chi _2) \cdot \cos \theta \\&\qquad +~\sqrt{v_1} \cdot \sum ^{\infty }_{m=2} \frac{1}{m!} \left( (h_m(\chi _1) + m \cdot h_{m-2}(\chi _1) ) \cdot \varphi (\chi _1) \cdot \right) \\&\qquad \cdot ~E\left[ \frac{\partial ^{m}}{\partial ^m X } \{ X \cdot \cos \theta + Y \cdot \sin \theta > \chi _2 \} \right] \\&\quad = \sqrt{v_1} \cdot \varphi (\chi _1) \cdot \overline{\Phi }(\chi _2) + \sqrt{v_1} \cdot \left( \overline{\Phi }(\chi _1) + \chi _1\right. \\&\qquad \left. \cdot ~\varphi (\chi _1) \right) \cdot \varphi (\chi _2) \cdot \cos \theta \\&\qquad + \sqrt{v_1} \cdot \varphi (\chi _1) \cdot \varphi (\chi _2) \cdot \sum ^{\infty }_{m=2} \frac{1}{m!} \left( h_m(\chi _1) \right. \\&\qquad \left. +~m \cdot h_{m-2}(\chi _1) \right) \cdot h_{m-1}(\chi _2) \cdot \cos ^m \theta \end{aligned}$$

For the next two covariance terms, we use the previous covariance term in the calculation.

$$\begin{aligned}&\mathrm {Cov}\left[ Q_1, \{ Q_1 > 0 \} \cdot \{ Q_2 \le 0 \} \right] \\&\quad =\mathrm {Cov}\left[ Q_1, \{ Q_1 > 0 \} \cdot (1 - \{ Q_2 > 0 \} ) \right] \\&\quad =\mathrm {Cov}\left[ Q_1, \{ Q_1 > 0 \} \right] - \mathrm {Cov}\left[ Q_1, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \mathrm {Cov}\left[ \sqrt{v_1} \cdot X, \{ X > \chi _1 \} \right] - \mathrm {Cov}\left[ \sqrt{v_1} \cdot X, \{ X > \chi _1 \}\right. \\&\qquad \left. \cdot ~\{ X \cdot \cos \theta + Y \cdot \sin \theta > \chi _2 \} \right] \\&\quad = \sqrt{v_1} \cdot \varphi ( \chi _1) - \sqrt{v_1} \cdot \mathrm {Cov}\left[ X, \{ X > \chi _1 \} \right. \\&\qquad \left. \cdot ~\{ X \cdot \cos \theta + Y \cdot \sin \theta > \chi _2 \} \right] \\&\quad = \sqrt{v_1} \cdot \varphi ( \chi _1) - \sqrt{v_1} \cdot \varphi (\chi _1) \cdot \overline{\Phi }(\chi _2)\\&\qquad - \sqrt{v_1} \cdot \left( \overline{\Phi }(\chi _1) + \chi _1 \cdot \varphi (\chi _1) \right) \cdot \varphi (\chi _2) \cdot \cos \theta \\&\qquad - \sqrt{v_1} \cdot \varphi (\chi _1) \cdot \varphi (\chi _2) \cdot \sum ^{\infty }_{m=2} \frac{1}{m!} \left( h_m(\chi _1)\right. \\&\qquad \left. +~m \cdot h_{m-2}(\chi _1) \right) \cdot h_{m-1}(\chi _2) \cdot \cos ^m \theta \\&\quad = \ \sqrt{v_1} \cdot \varphi (\chi _1) \cdot \Phi (\chi _2) - \sqrt{v_1} \cdot \left( \overline{\Phi }(\chi _1) + \chi _1 \cdot \varphi (\chi _1) \right) \\&\qquad \cdot ~\varphi (\chi _2) \cdot \cos \theta \\&\qquad - \sqrt{v_1} \cdot \varphi (\chi _1) \cdot \varphi (\chi _2) \cdot \sum ^{\infty }_{m=2} \frac{1}{m!} \left( h_m(\chi _1) \right. \\&\qquad \left. +~m \cdot h_{m-2}(\chi _1) \right) \cdot h_{m-1}(\chi _2) \cdot \cos ^m \theta \\&\mathrm {Cov}\left[ Q_1, \{ Q_1 \le 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad =\mathrm {Cov}\left[ Q_1, (1- \{ Q_1 > 0 \} ) \cdot \{ Q_2 > 0 \} \right] \\&\quad =\mathrm {Cov}\left[ Q_1, \{ Q_2 > 0 \} \right] - \mathrm {Cov}\left[ Q_1, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad =\mathrm {Cov}\left[ \sqrt{v_1} \cdot X, \{ Q_2 > 0 \} \right] - \mathrm {Cov}\left[ \sqrt{v_1} \cdot X, \{ Q_1 > 0 \} \right. \\&\left. \qquad \cdot ~\{ Q_2 > 0 \} \right] \\&\quad =\sqrt{v_1} \cdot \varphi (\chi _2) \cdot \cos \theta -\sqrt{v_1} \cdot \varphi (\chi _1) \cdot \overline{\Phi }(\chi _2) \\&\qquad - \sqrt{v_1} \cdot \left( \overline{\Phi }(\chi _1) + \chi _1 \cdot \varphi (\chi _1) \right) \cdot \varphi (\chi _2) \cdot \cos \theta \\&\qquad - \sqrt{v_1} \cdot \varphi (\chi _1) \cdot \varphi (\chi _2) \cdot \sum ^{\infty }_{m=2} \frac{1}{m!} \left( h_m(\chi _1) \right. \\&\qquad \left. +~m \cdot h_{m-2}(\chi _1) \right) \cdot h_{m-1}(\chi _2) \cdot \cos ^m \theta \end{aligned}$$

Now we begin the calculation of the covariance terms with respect to the second queue length. From the first line to the second we use the property that covariances are invariant to constants. Then, we use the Hermite polynomial expansion property and the Hermite polynomial generalization of Stein’s lemma once again.

$$\begin{aligned}&\mathrm {Cov}\left[ Q_2, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \mathrm {Cov}\left[ \sqrt{v_1} \cdot X, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_2} \cdot \mathrm {Cov}\left[ X \cdot \cos \theta + Y \cdot \sin \theta , \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_2} \cdot \cos \theta \cdot \mathrm {Cov}\left[ X, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\qquad + \sqrt{v_2} \cdot \sin \theta \cdot \mathrm {Cov}\left[ Y, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_2} \cdot \cos \theta \cdot \mathrm {Cov}\left[ X, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] + \sqrt{v_2} \\&\qquad \cdot \sin \theta \cdot \sum ^{\infty }_{m=0} \sum ^{\infty }_{n=0} \frac{1}{m!n!} E\left[ \frac{\partial ^{n+m}}{\partial ^m X \partial ^n Y} Y \cdot \{ Q_1 > 0 \} \right] \\&\qquad \cdot E\left[ \frac{\partial ^{n+m}}{\partial ^m X \partial ^n Y} \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_2} \cdot \cos \theta \cdot \mathrm {Cov}\left[ X, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\qquad + \sqrt{v_2} \cdot \sin \theta \cdot \sum ^{\infty }_{m=0} \frac{1}{m!} E\left[ \frac{\partial ^{1+m}}{\partial ^m X \partial Y} Y \cdot \{ Q_1 > 0 \} \right] \\&\qquad \cdot ~E\left[ \frac{\partial ^{1+m}}{\partial ^m X \partial Y} \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_2} \cdot \cos \theta \cdot \mathrm {Cov}\left[ X, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\qquad + \sqrt{v_2} \cdot \sin \theta \cdot \left( \overline{\Phi }(\chi _1) \cdot \varphi (\chi _2) \cdot \sin \theta \right) + \sqrt{v_2} \\&\qquad \cdot \sin \theta \cdot \left( \varphi (\chi _1) \cdot \chi _2 \cdot \varphi (\chi _2) \cdot \sin \theta \cdot \cos \theta \right) \\&\qquad + \sqrt{v_2} \cdot \sin \theta \cdot \sum ^{\infty }_{m=2} \frac{1}{m!} E\left[ \frac{\partial ^{m}}{\partial ^m X } \{ X > \chi _1 \} \right] \\&\qquad \cdot ~E\left[ \frac{\partial ^{m}}{\partial ^m X } \delta _{\chi _2}(X\cdot \cos \theta + Y\cdot \sin \theta ) \right] \\&\quad = \sqrt{v_2} \cdot \cos \theta \cdot \mathrm {Cov}\left[ X, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\qquad + \sqrt{v_2} \cdot \sin \theta \cdot \left( \overline{\Phi }(\chi _1) \cdot \varphi (\chi _2) \cdot \sin \theta \right) + \sqrt{v_2} \cdot \sin \theta \\&\qquad \cdot \left( \varphi (\chi _1) \cdot \chi _2 \cdot \varphi (\chi _2) \cdot \sin \theta \cdot \cos \theta \right) \\&\qquad + \sqrt{v_2} \cdot \sin \theta \cdot \sum ^{\infty }_{m=2} \frac{1}{m!} \varphi (\chi _1) \cdot h_{m-1}(\chi _1) \cdot \varphi (\chi _2) \\&\qquad \cdot ~h_{m}(\chi _2) \cdot \sin \theta \cdot \cos ^m \theta \end{aligned}$$

Lastly, for the next two covariance terms, we use the previous covariance term in the calculation.

$$\begin{aligned}&\mathrm {Cov}\left[ Q_2, \{ Q_1 > 0 \} \cdot \{ Q_2 \le 0 \} \right] \\&\quad = \mathrm {Cov}\left[ Q_2, \{ Q_1 > 0 \} \cdot (1 - \{ Q_2 > 0 \} ) \right] \\&\quad = \mathrm {Cov}\left[ Q_2, \{ Q_1 > 0 \} \right] - \mathrm {Cov}\left[ Q_2, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \}\right] \\&\quad = \sqrt{v_2} \cdot \mathrm {Cov}\left[ X \cdot \cos \theta + Y \cdot \sin \theta , \{ Q_1 > 0 \} \right] - \sqrt{v_2} \\&\qquad \cdot ~\mathrm {Cov}\left[ X \cdot \cos \theta + Y \cdot \sin \theta , \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_2} \cdot \cos \theta \cdot \varphi (\chi _1) - \sqrt{v_2} \cdot \mathrm {Cov}\left[ X \cdot \cos \theta \right. \\&\qquad \left. + Y \cdot \sin \theta , \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\mathrm {Cov}\left[ Q_2, \{ Q_1 \le 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \mathrm {Cov}\left[ Q_2, ( 1- \{ Q_1 > 0 \} ) \cdot \{ Q_2 > 0 \} \right] \\&\quad = \mathrm {Cov}\left[ Q_2, \{ Q_2 > 0 \} \right] - \mathrm {Cov}\left[ Q_2, \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_2} \cdot \mathrm {Cov}\left[ X \cdot \cos \theta + Y \cdot \sin \theta , \{ Q_2 > 0 \} \right] - \sqrt{v_2}\\&\qquad \cdot \, \mathrm {Cov}\left[ X \cdot \cos \theta + Y \cdot \sin \theta , \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \\&\quad = \sqrt{v_2} \cdot \varphi (\chi _2) - \sqrt{v_2} \cdot ~\mathrm {Cov}\left[ X \cdot \cos \theta + Y \right. \\&\qquad \left. \cdot \sin \theta , \{ Q_1 > 0 \} \cdot \{ Q_2 > 0 \} \right] \end{aligned}$$
Fig. 10
figure 10

Simulated probability of emptiness of queue 1 (Left). Simulated probability of emptiness of queue 2 (Right)

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Pender, J. An analysis of nonstationary coupled queues. Telecommun Syst 61, 823–838 (2016). https://doi.org/10.1007/s11235-015-0039-0

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