## Abstract

Wireless Sensor Networks (WSN) consist of a large number of sensors which have limited battery power. One of the major issues in WSN is the need to improve the overall network lifetime. Hence, WSN necessitate energy-efficient routing protocols. In this paper, a cross-layer routing protocol (PLOSA) is designed to offer a high delivery rate, a low end-to-end delay, and a low energy consumption. To achieve these goals, the transmission channel is divided into different slots, and a sensor has access to a slot related to its distance from the collector. The transmissions are then ordered within the frame from the farthest nodes to the closest ones which is a key point in order to ease forwarding and to conserve energy. We have conducted simulation-based evaluations to compare the performance of the proposed protocol against the framed aloha protocol. The performance results show that our protocol is a good candidate for WSN.

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## Appendix: Determination of function f

### Appendix: Determination of function *f*

In this appendix, we identify the best function *f* for different types of deployment of sensors in order to have a uniform distribution of packets on the slots of a frame.

For the sake of simplicity and without any loss of generality, we consider that sensors are distributed in a disk of radius 1 (we normalize all distances by the value of the radius). Let *ρ*(*u*) be the density of sensors at distance *u* (0<*u*≤1). The pdf of the distance is proportional to 2*ρ*(*u*)*u*
*d*
*u*. Let Γ(*r*) be the cdf of distance *r* between a sensor and the collector. We have thus

Let \(\overline {f}(x)=1-f(x)\). For a node at distance *r*, the path loss is given by *h*(*r*) with *h*(*r*)=*r*
^{α}/*k*=*L*
_{
m
a
x
}
*r*
^{α}. Note that both \(\overline {f}\) and *h* are increasing functions. A node at distance *r* has path loss *L*=*h*(*r*). With PLOSA, it computes *f*(*L*/*L*
_{
m
a
x
}) and then considers slot ⌊*S*
*f*(*L*/*L*
_{
m
a
x
})⌋ for the transmission. In order to equally spread the transmissions on all slots, our objective is thus to ensure that \(s= \overline {f}(L/L_{max})\) is a uniformly distributed random variable on interval [0,1].

In the following, we denote by *F*
_{
X
}(*x*) the cdf of r.v. *X*. For any random variable *X* and any increasing function *g*, the cdf of *Y*=*g*(*x*) is given by *F*
_{
Y
}(*y*)=*F*
_{
X
}(*g*
^{−1}(*y*)). The cdf of *s* is thus

We search *f* such that *F*
_{
s
}(*s*)=*s* (uniform distribution). Hence, we choose

We know determine function Γ(*r*) with multi-hop transmission. Nodes close to the sink transmit many more packets as they have to transmit both their own data and the packets they forward. We are not interested to know which particular node transmits a given packet. We then consider that packets are forwarded by virtual nodes (one virtual node is added for each retransmitted packet) and compute the density of virtual and real nodes. This density is thus higher close to the collector. We assume a uniform distribution *ρ* of real nodes.

The average transmission distance of a node is assumed to be equal to *β* with *β*<1. The unit disk is divided in *N* rings of length *β* plus one central disk of radius *α* (see Fig. 10). Let ring *i* (1≤*i*≤*N*) be defined by the area between circle of radius *α*+*β*(*i*−1) and circle of radius *α*+*β*
*i*. We have *α*+*β*
*N*=1 and thus *β*=(1−*α*)/*N*. Let *A*
_{
i
} be the area of ring *i*. We have *A*
_{
i
}=*π*(*α*+*β*
*i*)^{2}−*π*(*α*+*β*(*i*−1))^{2}. We deduce

If we consider that sensors in ring *i* forward all the packets that are sent by sensors in ring *i*+1 and that these latter packets include also forwarded packets from higher rings, we can easily state that the load in ring *i* is equal to the fresh traffic generated on the area from ring *i* till ring *N*. The total packet arrival rate Λ_{
i
} in ring *i* is this:

where *λ* is the arrival rate of new packets for a node. After some elementary computation, we find

The total density of real and virtual sensors is given by

Equation 7 gives discrete values, but we want to manage a continuous equation as in Eq. 1 . We fit the curve at the middle of each ring that is for *r*=*α*+*β*(*i*−1/2). We can then substitute *i* by \(\frac { 2 r - (2 \alpha - \beta )}{2 \beta }\). Equation 7 becomes thus

if *r*≥*α*−*β*/2. After some elementary computations, it comes

As *r* < 1 and *β* < 1, we have

By combining Eqs. 10 and 1, we find

Using Eq. 3, we easily find function *f*

or in other words for a node with path loss *L* it is necessary to compute *f*(*L*/*L*
_{
m
a
x
})=1−(*L*/*L*
_{
m
a
x
})^{1/α}.

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Espes, D., Lagrange, X. & Suárez, L. A cross-layer MAC and routing protocol based on slotted aloha for wireless sensor networks.
*Ann. Telecommun.* **70**, 159–169 (2015). https://doi.org/10.1007/s12243-014-0433-8

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DOI: https://doi.org/10.1007/s12243-014-0433-8