A simple analytical throughput–delay model for clustered FiWi networks

Abstract

A fiber-wireless (FiWi) network integrates a passive optical network (PON) with wireless mesh networks (WMNs) to provide high-speed backhaul via the PON while offering the flexibility and mobility of a WMN. Generally, increasing the size of a WMN leads to higher wireless interference and longer packet delays. We examine the partitioning of a large WMN into several smaller WMN clusters, whereby each cluster is served by an optical network unit (ONU) of the PON. Existing WMN throughput–delay analysis techniques considering the mean load of the nodes at a given hop distance from a gateway (ONU) are unsuitable for the heterogeneous nodal traffic loads arising from clustering. We introduce a simple analytical queuing model that considers the individual node loads to accurately characterize the throughput–delay performance of a clustered FiWi network. We verify the accuracy of the model through extensive simulations. We employ the model to examine the impact of the number of clusters on the network throughput–delay performance. We find that with sufficient PON bandwidth, clustering substantially improves the FiWi network throughput–delay performance.

Appendices

Appendix 1: Review of $M/M/1/K$ Queue

Define the traffic intensity as $\rho = \lambda / \mu$, where $\lambda$ and $\mu$ are the packet arrival rate and the packet service rate of the queue holding at most $K$ packets. The queue holds $K$ packets, i.e., blocks newly arriving packets, with probability [30]:

$$\begin{aligned} P_{\mathrm{M},K}(\rho ,K) = \left\{ \begin{array}{ll} \frac{(1-\rho )\rho ^K}{1-\rho ^{K+1}} &{} \text{ if } \quad \rho \ne 1 \\ \frac{1}{K+1} &{} \text{ if } \quad \rho = 1. \end{array} \right. \end{aligned}$$

(29)

The probability of the queue being empty is:

$$\begin{aligned} P_{\mathrm{M},0}(\rho ,K) = \left\{ \begin{array}{ll} \frac{1-\rho }{1-\rho ^{K+1}} &{} \text{ if } \quad \rho \ne 1 \\ \frac{1}{K+1} &{} \text{ if } \quad \rho = 1 \end{array} \right. \end{aligned}$$

(30)

and $P_{0,i}$ in Eq. (5) can be obtained as:

$$\begin{aligned} P_{0,i}=P_{\mathrm{M},0}(\rho _i,K). \end{aligned}$$

(31)

The average queue length is [30]:

$$\begin{aligned} L_\mathrm{M}(\rho ,K) = \left\{ \begin{array}{ll} \frac{\rho }{1-\rho }-\frac{\rho (K\rho ^K+1)}{1-\rho ^{K+1}} &{} \text{ if } \quad \rho \ne 1 \\ \frac{K(K-1)}{2(K+1)} &{} \text{ if } \quad \rho = 1. \end{array} \right. \end{aligned}$$

(32)

The average waiting time is:

$$\begin{aligned} W_\mathrm{M}(\mu ,\lambda ,K)=\frac{1}{\mu } +\frac{L_{\mathrm{M}}(\rho ,K)}{\lambda [1-P_{\mathrm{M},K}(\rho ,K)]}. \end{aligned}$$

(33)

Appendix 2: Review of $M/D/1/K$ Queue

Define input packet rate $\lambda$, output packet rate $\mu$, and traffic intensity $\rho =\lambda /\mu$. Denote $P_{\mathrm{D}, k}(\rho , K),\ k=0, \ldots , K$, for the stationary state probabilities of holding $k$ packets in the queue. For $0 \le k \le K-1$, the steady state probability can be obtained with the recursion [30]:

$$\begin{aligned} P_{\mathrm{D}, k}(\rho , K)=\lambda a_{k-1} P_{\mathrm{D},0}(\rho , K)+\lambda \sum _{j=1}^k a_{k-j} P_{\mathrm{D},j}(\rho , K), \end{aligned}$$

(34)

where $a_n=\frac{1}{\lambda } (1-\sum _{j=1}^n{\mathrm{e}^{-\rho }\rho ^j/j!})$. The $K$th state probability, i.e., the blocking probability, is:

$$\begin{aligned} P_{\mathrm{D}, K}(\rho , K)= \rho P_{\mathrm{D}, 0}(\rho , K) -(1-\rho )\sum _{j=1}^{K-1} P_{\mathrm{D}, j}(\rho , K).\nonumber \\ \end{aligned}$$

(35)

The recursion starts with $P_{\mathrm{D},0}=1$ and the state probabilities are normalized with the equation $\sum _{i=0}^K P_{\mathrm{D},i}(\rho , K)=1$. An explicit formula for $P_{\mathrm{D}, k}(\rho , K)$ is derived in [14], but the calculation process involves a large number operations for large $K$ and may not be suitable for computational work [74]. With the state probabilities, the average waiting time $W_\mathrm{D}(\mu , \lambda , K)$ of an $M/D/1/K$ queue can be evaluated by applying Little’s law:

$$\begin{aligned} W_\mathrm{D}(\mu , \lambda , K)=\frac{1}{\mu }+ \frac{L_\mathrm{D}(\rho , K)}{\lambda [1-P_{\mathrm{D}, K}(\rho , K)]}, \end{aligned}$$

(36)

where $L_\mathrm{D}(\rho , K)=\sum _{k=0}^K k P_{\mathrm{D},k}(\rho , K)$ is the average length of the $M/D/1/K$ queue.

Appendix 3: Bandwidth Fair Sharing for WMN

One of the major problems of a WMN is the fairness share problem [28, 41] where the nodes with higher hop distance suffer from lower throughput compared to the nodes with lower hop distance. For a TDMA system, it is desired that the wireless nodes closer to the gateways should be allocated more radio resources, i.e., higher channel access probability $p(x)$ for lower hop count $x$, since they have to provide more relay services. If the scheduling scheme failed to provide sufficient radio resources to the wireless nodes closer to the gateways to maintain a reasonable relay traffic intensity, then low throughout and high delay would occur due to frequent buffer overflow and further affect the overall performance of the WMN. Liu and Liao [50] proposed the following wireless channel allocation scheme which we apply to the FiWi network:

$$\begin{aligned} \frac{p(x)}{p(x+1)}=N_\mathrm{r}(x) \left[ 1 + \frac{1}{R(x)} \right] ,\ x=1, 2, \ldots , H-1.\nonumber \\ \end{aligned}$$

(37)

where

$$\begin{aligned} R(x)=\sum _{i=x}^H\prod _{j=x}^iN_\mathrm{r}(i),\quad x=1, 2, \ldots , H-1, \end{aligned}$$

and

$$\begin{aligned} N_\mathrm{r}(x) = \left\{ \begin{array}{ll} \frac{N(x+1)}{N(x)} &{} \text{ if } \quad x=1, \ldots , H-1 \\ 0 &{} \text{ if } \quad x = H. \end{array} \right. \end{aligned}$$

(38)

Equation (37) gives the $p(x)$ design criteria which provide fair throughput to all wireless mesh nodes regardless of the hop distances under the assumption that the relayed traffic is distributed evenly among the wireless mesh nodes. Inequality

$$\begin{aligned} q(x)>1- \frac{1}{1+R(x)}. \end{aligned}$$

(39)

specifies a lower bound for the forwarding probability $q(x)$ ensuring that an $x$-hop node is capable of providing fair bandwidth allocation [50].