Monotonicity Properties for Multi-Class Queueing Systems

Abstract

We study multi-dimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service processes, we give sufficient conditions in order to compare sample-path wise the workload and the number of users under different policies. This allows us to evaluate the performance of the system under various policies in terms of stability, the mean overall delay and the mean holding cost. We apply the general framework to linear networks, where users of one class require service from several shared resources simultaneously. For the important family of weighted α-fair policies, stability results are derived and monotonicity of the mean holding cost with respect to the fairness parameter α and the relative weights is established. In order to broaden the comparison results, we investigate a heavy-traffic regime and perform numerical experiments. In addition, we study a single-server queue with two user classes, and show that under Discriminatory Processor Sharing (DPS) or Generalized Processor Sharing (GPS) the mean overall sojourn time is monotone with respect to the ratio of the weights. Finally we extend the framework to obtain comparison results that cover the single-server queue with an arbitrary number of classes as well.

Appendices

Appendix A: Proof of Lemma 5.1

For a given state $\vec{n}$, the α-fair allocation is the vector (s ₀,s ₁,..., s _L ) that solves the optimization problem (7). If n _i  > 0, then s _i  = C _i  − s ₀. The objective function in Eq. 7 expressed in terms of the value s ₀, is concave in s ₀. Taking the derivative of Eq. 7 with respect to s ₀ and setting it equal to zero, we obtain that $s_0^{\pi(\alpha,w)}(\vec{n})$ satisfies

$$ w_0\cdot n_0^\alpha \cdot (s_0^{\pi(\alpha, w)}(\vec{n}))^{-\alpha} = \sum\limits_{i=1}^L w_i\cdot n_i^\alpha \cdot (C_i-s_0^{\pi(\alpha, w)}(\vec{n}))^{-\alpha}, $$

or equivalently

$$ 1=\sum\limits_{i=1}^L\frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0} \frac{s_0^{\pi(\alpha, w)}(\vec{n})}{C_i-s_0^{\pi(\alpha, w)}(\vec{n})}\Bigr)^\alpha. $$

(28)

The function $\sum_{i=1}^L \frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0}{C_i-s_0}\Bigr)^\alpha$ is non-decreasing in s ₀. Hence, when either n _i or $\frac{w_i}{w_0}$ increases, by equality (28) the corresponding value of s ₀ must decrease. Statements (i) and (iii) follow now immediately.

Statement (ii) follows similarly by noting that

$$\begin{array}{lll} \Bigl(\sum\limits_{i=1}^L \frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0^{\pi(\gamma,w)}(\vec{n})}{C_i-s^{\pi(\gamma,w)}_0(\vec{n})}\Bigr)^\gamma \Bigl)^{\frac{1}{\gamma}} = 1&=& \Bigl(\sum\limits_{i=1}^L \frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0^{\pi(\beta,w)}(\vec{n})}{C_i-s^{\pi(\beta,w)}_0(\vec{n})}\Bigr)^\beta \Bigl)^{\frac{1}{\beta}} \\[4pt] &=&\Bigr(\sum\limits_{i=1}^L\frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0^{\pi(\beta,w)}(\vec{n})}{C_i-s^{\pi(\beta,w)}_0(\vec{n})}\Bigr)^\beta \Bigl)^{\frac{r}{r\beta}}\\ & \geq& \Bigr(\sum\limits_{i=1}^L\frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0^{\pi(\beta,w)}(\vec{n})}{C_i-s^{\pi(\beta,w)}_0(\vec{n})}\Bigr)^{r\beta} \Bigl)^{\frac{1}{r\beta}} \\&=& \Bigr(\sum\limits_{i=1}^L\frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0^{\pi(\beta,w)}(\vec{n})}{C_i-s_0^{\pi(\beta,w)}(\vec{n})}\Bigr)^\gamma \Bigl)^{\frac{1}{\gamma}}, \end{array}$$

with rβ = γ and r > 1. Hence $s^{\pi(\beta,w)}_0(\vec{n})\leq s^{\pi(\gamma,w)}_0(\vec{n})$.□

Appendix B: Proof of Proposition 5.6

By the conjecture of Kang et al. (2009), the scaled workload in a node is independent of α. In addition, it is stated that the diffusion scaled number of users, $\vec{\hat N}^{k,\pi(\alpha)}(t)$, converges in distribution as k→ ∞ to $\vec{\hat N}^{\pi(\alpha)}(t)=\Delta(\vec{\hat{V}}^{\pi(\alpha)}(t))$, where the lifting mapping $\Delta: \mathbf{R}_+^2 \to \mathbf{R}_+^3$ is as defined in Kang et al. (2009) and Kelly and Williams (2004). This is equivalent to saying that there are q ₁(α), q ₂(α) ≥ 0 such that

$$ \label{eq:dif_n} \hat N_i^{\pi(\alpha)} = \rho_iq_i(\alpha)^{\frac{1}{\alpha}}, \ \ i=1,2 \ \ \mbox{ and } \ \ \hat N_0^{\pi(\alpha)}=\rho_0(q_1(\alpha)+q_2(\alpha))^{\frac{1}{\alpha}}. $$

(29)

Using this representation for the number of users, we can describe the effect the parameter α has on the holding cost.

We compare the holding cost under two α-fair policies with parameters α ₁ and α ₂ for a given workload in both nodes. So we have at each time that the workload in a node is $\hat V_i^{\pi(\alpha)}=\hat v_i,$ independent of α, i = 1,2 (from now on we will drop the dependence on t). Using the representation as in Eq. 9 and the fact that ρ ₀ + ρ _i  = C _i , this gives

$$ (C_i-\rho_0)q_i(\alpha)^{1/\alpha}+\rho_0\frac{\mu_i}{\mu_0}(q_1(\alpha)+q_2(\alpha))^{1/\alpha} =\hat v_i. $$

(30)

Together with Eq. 29, the holding cost for an α-fair policy can be written as

$$\begin{array}{lll} \sum\limits_{i=0}^2 c_i \hat N_i^{\pi(\alpha)} &=& c_0\rho_0(q_1(\alpha)+q_2(\alpha))^{\frac{1}{\alpha}} +c_1(C_1-\rho_0) q_1(\alpha)^{\frac{1}{\alpha}} +c_2(C_2-\rho_0)q_2(\alpha)^{\frac{1}{\alpha}} \\ &=& c_1 \Bigl( (C_1-\rho_0)q_1(\alpha)^{1/\alpha}+\rho_0\frac{\mu_1}{\mu_0}(q_1(\alpha)+q_2(\alpha))^{1/\alpha}\Bigr) \\ && +\,c_2\Bigl((C_2-\rho_0)q_2(\alpha)^{1/\alpha}+\rho_0\frac{\mu_2}{\mu_0}(q_1(\alpha)+q_2(\alpha))^{1/\alpha}\Bigr) \\ && +\, \frac{c_0\mu_0-c_1\mu_1-c_2\mu_2}{\mu_0}\rho_0(q_1(\alpha)+q_2(\alpha))^{\frac{1}{\alpha}} \\ &\stackrel{d}{=}& c_1 \hat v_1+ c_2\hat v_2 +\frac{c_0\mu_0-c_1\mu_1-c_2\mu_2}{\mu_0}\rho_0(1+f(\alpha)^\alpha)^{\frac{1}{\alpha}}q_2(\alpha)^{\frac{1}{\alpha}}, \end{array}$$

(31)

where $f(\alpha):=\Bigl(\frac{q_1(\alpha)}{q_2(\alpha)}\Bigr)^\frac{1}{\alpha}$.

For i = 2, Eq. 30 gives

$$\begin{array}{lll} &&{\kern-6pt} q_2(\alpha_1)^{1/\alpha_1}\Bigl(C_2-\rho_0+\rho_0\frac{\mu_2}{\mu_0}(1+f(\alpha_1)^{\alpha_1})^{1/\alpha_1}\Bigr)\\ &&{\kern4pt} =q_2(\alpha_2)^{1/\alpha_2}\Bigl(C_2-\rho_0+\rho_0\frac{\mu_2}{\mu_0}(1+f(\alpha_2)^{\alpha_2})^{1/\alpha_2}\Bigr). \label{i2} \end{array}$$

(32)

From Eq. 32 we conclude that

$$ (1+f(\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}q_2(\alpha_1)^{\frac{1}{\alpha_1}} < \ (=) \ (1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}}q_2(\alpha_2)^{\frac{1}{\alpha_2}} $$

(33)

if and only if

$$\frac{(1+f(\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}}{(C_2-\rho_0+\rho_0\frac{\mu_2}{\mu_0}(1+f(\alpha_1)^{\alpha_1})^{1/\alpha_1}} < \ (=) \ \frac{(1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}}}{(C_2-\rho_0+\rho_0\frac{\mu_2}{\mu_0}(1+f(\alpha_2)^{\alpha_2})^{1/\alpha_2}}, $$

if and only if

$$ (1+f(\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}< \ (=) \ (1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}}. $$

(34)

Let b be such that $\hat v_1=b\hat v_2$. Assume without loss of generality $\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0} \leq b\leq 1$. (For states with b > 1 the analysis is the same, with only the roles of nodes 1 and 2 interchanged.) Note that when $b=\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}$ we are on the edge of the cone as described in Eq. 10. In Lemma 1 (see below) we prove that $(1+f(\alpha)^{\alpha})^{\frac{1}{\alpha}}$ is indeed strictly decreasing in α when $\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}< b \leq 1$ and is constant when $b=\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}$, the edge of the cone. Assuming the probability mass is not all concentrated on the edge of the cone, we conclude from Eq. 31 and the equivalence between Eqs. 33 and 34, that the mean holding cost is strictly decreasing (strictly increasing) in α when c ₁ μ ₁ + c ₂ μ ₂ < c ₀ μ ₀ (c ₁ μ ₁ + c ₂ μ ₂ > c ₀ μ ₀). □

The following lemma is used in the proof of Proposition 5.6.

Lemma 1

The function (1 + f(α)^α)^1/α with $f(\alpha)=\Bigl(\frac{q_1(\alpha)}{q_2(\alpha)}\Bigr)^{\frac{1}{\alpha}}$, is strictly decreasing in α when $\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0} < b\leq 1$ and is constant when $b=\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}.$ Here b satisfies $\hat v_1=b\hat v_2$.

Proof

From $\hat v_1=b\hat v_2$ and Eq. 30 we obtain the relation

$$ (C_1-\rho_0)q_1(\alpha_i)^{\frac{1}{\alpha_i}}+(1-b)\rho_0\frac{\mu_1}{\mu_0}(q_1(\alpha_i)+q_2(\alpha_i))^{\frac{1}{\alpha_i}}=b(C_2-\rho_0)\frac{\mu_1}{\mu_2}q_2(\alpha_i)^{\frac{1}{\alpha_i}}, $$

hence when we divide both sides by $q_2(\alpha_i)^{\frac{1}{\alpha_i}}$, we obtain

$$ (C_1-\rho_0)f(\alpha_i)+(1-b)\rho_0\frac{\mu_1}{\mu_0}(1+f(\alpha_i)^{\alpha_i})^{\frac{1}{\alpha_i}}=b(C_2-\rho_0)\frac{\mu_1}{\mu_2}. $$

(35)

By Eq. 35 we have that f(α) = 0 if and only if $b=\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}$.

Assume $b=\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}$. Then f(α) = 0 and hence the function $(1+f(\alpha)^{\alpha})^{\frac{1}{\alpha}}$ is constant.

Now assume $\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0} < b\leq 1$. So f(α) > 0 for all α. Take α ₁ < α ₂ and let r > 1 be such that α ₂ = rα ₁. Then

$$(1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}} = (1^{r\alpha_1}+f(r\alpha_1)^{r\alpha_1})^{\frac{1}{r\alpha_1}} <(1^{\alpha_1}+f(r\alpha_1)^{\alpha_1})^{\frac{r}{r\alpha_1}} =(1+f(r\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}, $$

(36)

since 1 + f(rα ₁) > 1. Suppose f(α ₂) = f(rα ₁) ≤ f(α ₁). From Eq. 36, we then obtain $(1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}}< (1+f(\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}$. However, from Eq. 35 we know that when f(α ₂) ≤ f(α ₁), then $(1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}}\geq (1+f(\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}$, hence we have a contradiction. So we conclude that f(α ₂) > f(α ₁), and hence f(α) is strictly increasing in α and from Eq. 35 it then follows that $(1+f(\alpha)^\alpha)^\frac{1}{\alpha}$ is strictly decreasing in α.□