Page 118 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 1

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 1

under jamming of a transmitter with a group of receivers We say that the communication from the transmitter to
when the channels are affected by Rayleigh fading is mod‑ the receiver is maintained if and only if the SINR at the
eled as a zero‑sum power power allocation game. receiver is greater than or equal to a given TCC , i.e., the
(2) Existence and uniqueness of the equilibrium in power following condition holds:
allocation strategies are proven. Thus, in contrast to
Colonel Blotto games, if channels are affected by Rayleigh SINR( , ) ≥ . (2)
fading, then stability of communication connectivity in a This TCC depends on the system’s requirements, such as
multi‑link system can be maintained without introducing bit rate and bit error rate (BER).
a random factor for a decision maker. Then, since and are random variables, the probability
(3) We reduce the problem of inding the equilibrium in that the link between the transmitter and the receiver is
power resource allocation strategies to the problem of maintained, i.e., PCC, is given by
solving a ixed point equation in a scalar variable. An al‑
gorithm based on the bisection method to ind the ixed ( , ) = (SINR( , ) ≥ ) . (3)
point (and so equilibrium strategies) is developed, and its
convergence is proven. This algorithm can be considered For the case when the channels are affected by Rayleigh
to be a learning algorithm since it allows one to reduce the fading (i.e., and are exponential random variables
zone of uncertainty for the equilibrium by a factor of two with means [ ] = and [ ] = ℎ, respectively), by
on iteration. [14, 15], the probability (3) can be represented as follows:
The organization of this paper is as follows. In Section 2,
a short summary of an SLCC model is given. In Section 3, − ℎ
its generalization for the MLCC scenario is suggested as a ( , ) = . (4)
zero‑sum game between the transmitter communicating 1 +
with a group of receivers and the jammer. In Section 4, the ℎ
equilibrium strategies are designed and the uniqueness Note that ( , ) is continuous in ≥ 0 and ≥ 0 and
of the equilibrium is proven. In Section 5, the equilibrium (0, ) = 0 for ≥ 0.
strategies are established in closed form for the bound‑
ary cases of network parameters: (a) for small or large 3. A MULTI‑LINK COMMUNICATION CON‑
TCC, (b) for small or large total jamming and transmission NECTIVITY MODEL
power budgets and (c) for small background noise at the
receivers. In Section 6, an algorithm to design the equilib‑ In this section, we generalize the SLCC problem to the
rium in the general case is presented and its convergence MLCC problem involving a transmitter with receivers as
is proven. Finally, in Section 7, illustrations of the results follows:
are given, and, in Section 8, conclusions are offered. All
proofs are provided in the appendix. • We assume that the transmitter is equipped with
directed antennas to communicate with receivers.

2. SHORT OVERVIEW OF A SINGLE‑LINK • Let = ( , … , ) be the strategy of the transmitter

1
COMMUNICATION CONNECTIVITY where is the power assigned to communicate with

MODEL the receiver and
In this section following [14, 15] we give a short overview = and ≥ 0, ∈ (5)

of the SLCC model with one transmitter communicating ∈
directly to a single receiver. This communication is af‑
fected by hostile interference from a jammer. Let and with is the transmitter’s total power budget and
be the channel power gains from the transmitter to the ≜ {1, … , }. Denote by the set of all feasible
receiver and the jammer to the receiver, respectively. In strategies for the transmitter.
practice, the channel power fading gains depend on dis‑ • Let the jammer also be equipped with directed an‑
tances, fading and antenna characteristics. In general, the tennas.
channel power gains are random variables (e.g., repre‑
sentingchannelfading)withmeans [ ] = and [ ] = • Let = ( , … , ) be strategy of the jammer, where

1
ℎ. Let and be the power levels used by the transmitter is the power assigned to jam the communication

and the jammer, respectively. Thus, ℝ is the set of fea‑ between the transmitter with receiver , and
+
sible strategies for the transmitter as well as for the jam‑
mer. The receiver also is affected by noise power . Thus, = and ≥ 0, ∈ (6)

the SINR at the receiver is given by ∈
with is the total jamming power budget. Denote by
SINR( , ) = . (1)
+ the set of all feasible strategies for the jammer.

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