Performance Analysis of Opportunistic Schedulers Using SPNs

Abstract

In Chap. 2, we have introduced the model decomposition and iteration technique in SPNs to deal with the state space explosion problem. In this chapter, we adopt this technique to study the performance of wireless opportunistic schedulers in multiuser systems under a dynamic data arrival setting. We first develop a framework based on Markov queueing model and then analyze it by applying the decomposition and iteration technique. Since the state space size in our analytical model is small, the proposed framework shows an improved efficiency in computational complexity. Based on the established analytical model, performance of both opportunistic and non-opportunistic schedulers are studied and compared in terms of average queue length, mean throughput, average delay and dropping probability. Analytical results demonstrate that the multiuser diversity effect as observed in the infinite backlog scenario is only valid in the heavy traffic regime. The performance of the Channel-Aware (CA) opportunistic schedulers is worse than that of the non-opportunistic round robin scheduler in the light traffic regime, and becomes worse especially with the increase of the number of users. Simulations are also performed to verify the accuracy of the analytical results.

Appendices

Appendix 1: Determination of $p_{l,m}^{n}$ in Rayleigh Fading Channel

For Rayleigh fading channel, $p_{l,m}^{n}$ can be determined as follows [12]. Assume the state transitions of the FSMC happen only between adjacent states, i.e.

$$\displaystyle{ p_{l,m}^{n} = 0,\quad \vert l - m\vert \geq 2. }$$

(3.35)

Let γ _n, l , $(l = 1,\ldots,L - 1)$, denotes the SNR threshold value between the l-th and (l + 1)-th states of the FSMC model for user n. The adjacent-state transition probability can be calculated as

$$\displaystyle{ p_{l,l+1}^{n} = \frac{\chi (\gamma _{n,l+1})\Delta T} {\pi _{n,l}},\quad l = 1,\ldots,L - 1, }$$

(3.36)

$$\displaystyle{ p_{l,l-1}^{n} = \frac{\chi (\gamma _{n,l})\Delta T} {\pi _{n,l}},\quad l = 2,\ldots,L. }$$

(3.37)

Here, χ(γ _n ) denotes the level cross rate at an instantaneous SNR γ _n and is given by

$$\displaystyle{ \chi (\gamma _{n}) = \sqrt{\frac{2\pi \gamma _{n } } {\overline{\gamma }}} f_{d}^{n}\exp (-\frac{\gamma _{n}} {\overline{\gamma }_{n}}), }$$

(3.38)

where $f_{d}^{n}$ denotes the mobility-induced Doppler spread, $\overline{\gamma }_{n} = \mathbb{E}[\gamma _{n}]$ is the average received SNR, and $\pi _{n,l}(l \in \mathcal{L})$ denotes the stationary probability that the FSMC is in state l given by

$$\displaystyle{ \pi _{n,l} =\exp (\gamma _{n,l}/\overline{\gamma }_{n}) -\exp (\gamma _{n,l+1}/\overline{\gamma }_{n}). }$$

(3.39)

Finally, $p_{l,l}^{n}$ can be derived from the normalizing condition $\sum _{m=1}^{L}p_{l,m}^{n} = 1$ as

$$\displaystyle{ p_{l,l}^{n} = \left \{\begin{array}{ll} 1 - p_{l,l+1}^{n} - p_{l,l-1}^{n},&(l = 2,\ldots,L - 1), \\ 1 - p_{l,l+1}^{n}, &(l = 1), \\ 1 - p_{l,l-1}^{n}, &(l = L). \end{array} \right. }$$

(3.40)

Appendix 2: Convergence of the Fixed Point Iteration

According to Sect. 2.2, in order to prove the convergence of the fixed point iteration for the decomposed DSPN model as described in (3.34), it is sufficient to show that the following lemma is true.

Lemma 3.1.

The embedded Markov chain $(\mathcal{H}_{n,t},\mathcal{Q}_{n,t})$ for the n-th subsystem is irreducible, if $K \leq R_{n,L}\Delta T$ .

Proof.

It has been proved in [7] that the Markov chain for the single user system is irreducible. Similarly, we prove Lemma 3.1 by showing that for each transition from state (l, k) to (m, h), there exists a multi-transition path $(l,k) \rightarrow (l^{{\ast}},k) \rightarrow (l^{{\ast}},h) \rightarrow (m,h)$ with non-zero probability, where $R_{n,l^{{\ast}}}\Delta T \geq k$. Since $K \leq R_{n,L}\Delta T$, there always exists such l ^∗ that satisfies this condition.

Since the FSMC model is irreducible, we have that $p_{l,l^{{\ast}}}^{n}$, $p_{l^{{\ast}},l^{{\ast}}}^{n}$ and $p_{l^{{\ast}},m}^{n}$ are all positive. Now we shall verify the following inequalities:

• (a) $\nu _{k,k}^{n,l}\tilde{g}_{n}(\mathbf{M}) +\nu _{ k,k}^{n,0}(1 -\tilde{g}_{n}(\mathbf{M})) > 0$;

• (b) $\nu _{k,h}^{n,l^{{\ast}} }\tilde{g}_{n}(\mathbf{M}) +\nu _{ k,h}^{n,0}(1 -\tilde{g}_{n}(\mathbf{M})) > 0$;

• (c) $\nu _{h,h}^{n,l^{{\ast}} }\tilde{g}_{n}(\mathbf{M}) +\nu _{ h,h}^{n,0}(1 -\tilde{g}_{n}(\mathbf{M})) > 0$.

For inequality 1), since $A_{n,t} = k -\max [0,k - R_{n,l}\Delta T] \geq 0$ (R _n, 0 = 0 included), we have $\nu _{k,k}^{n,l} > 0$ and $\nu _{k,k}^{n,0} > 0$. Therefore, inequality 1) is true with $\tilde{g}_{n}(\mathbf{M}) \in [0,1]$. The proof of inequality 3) is similar.

For inequality 2), since $A_{n,t} = h -\max [0,k - R_{n,l^{{\ast}}}\Delta T] \geq 0$, we have $\nu _{k,h}^{n,l^{{\ast}} } > 0$, where $R_{n,l^{{\ast}}}\Delta T \geq k$. Now consider both the cases when the value of k is zero or not. If k = 0, we have $\tilde{g}_{n}(\mathbf{M}) = 0$ and $\nu _{k,h}^{n,0} > 0$; otherwise, if k > 0, we have $\tilde{g}_{n}(\mathbf{M}) > 0$ and $\nu _{k,h}^{n,0} \geq 0$. Thus, the inequality 2) is true under both cases.

According to (3.29), we have $p_{(l,k),(l^{{\ast}},k)}^{n} > 0$, $p_{(l^{{\ast}},k),(l^{{\ast}},h)}^{n} > 0$ and $p_{(l^{{\ast}},h),(m,h)}^{n} > 0$, where $R_{n,l^{{\ast}}}\Delta T \geq k$, which prove the existence of the multi-transition path with non-zero probability for each transition from state (l, k) to (m, h).