Page 83 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 6 – Wireless communication systems in beyond 5G era

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 6

where denotes the outcome of an iteration of a simu‑ 8. COMPLEXITY ANALYSIS

lation and is the probability of occurrence for the out‑ In this section, we embark on the complexity analysis of

come . The values of probabilities , , , ⋯, are the proposed algorithms described in Section 7. Further‑
2

1
3

such that [87]:
more, we approach the complexity analysis from the time
> 0, ∀ ∈ (42) complexity approach. The time complexity analysis de‑

picts the worst‑case running performance time of an al‑
gorithm. We characterise the complexity of the proposed
= 1, ∀ ∈ (43) algorithms in this work with the big‑ . The big‑ in‑

∈ dicates the upper‑bound execution in terms of time (or
Therefore, the expectation of the outcomes in is given memory)ofanalgorithm[89, 90]. Weproceedbydescrib‑
as ing the computational complexity of the multi‑tier multi‑
( ) = , ∀ ∈ (44) domain slice user association algorithms. Moreover, we

discuss the computational complexity of distributed re‑
∈
cursive backtracking for multiplayer multi‑domain game
Assume that from the values obtained in , the number algorithms, and lastly, the M‑TTSD resource allocation.
occurs times, the number occurs times, the
2
1
1
2
number occurs times, and so on. Hence, from the 8.1 Computational complexity of the multi‑
3
3
laws of large numbers, we have
tier multi‑domain slice user association
= + + ⋯ + (45) We describe the computational complexity of the multi‑

2 2

1 1
∈ tier multi‑domain slice user as ( ⋅ |ℐ|). Where is the
Therefore, number of tiers in the multi‑tier network in a domain
and |ℐ| denotes the cardinality of the set of InP. Conse‑
1 1 2
= 1 + 2 + ⋯ + (46) quently, the computational complexity of the association

algorithm is dependent on the number of tiers and do‑
∈
main networks.

With beingalargenumber, thefrequency ofthevalue

approaches its probability such that: 8.2 Computational complexity of the dis‑

≈ . (47) tributed recursive backtracking multi‑
player multi‑domain
Hence, substituting (47) into (44), we have The complexity of the distributed recursive backtrack‑
ing multiplayer multi‑domain game algorithm is given by
1
( ) = = (48) (|ℐ| ⋅ (| | + | |)). It is important to note that in sce‑

narios where | | ≫ | |, its big‑ can be approximated
∈ ∈
as (|ℐ| ⋅ | |). However where | | ≫ | |, then, big‑ is
The expression in ((48)) is employed in the determination approximated as (|ℐ| ⋅ | |).
of the variance of given as:
2
( ) = ( ) − ( ( )) 2 (49) 8.3 Computational complexity of the multi‑
tier multi‑tenant multi‑slice multi‑domain
Algorithm 3 Monte Carlo simulation resource allocation
Input: , , , , , , , , , : ∈ ℐ, ∈ , ∈ ℋ In the similitude of subsections 8.1 and 8.2, the big‑ for
,
1: for ← 1 to do the multi‑tier multi‑tenant multi‑slice multi‑domain re‑
2: Generate the random variables of the network source allocation is given by ( ⋅|ℐ|⋅| |⋅| |). Where

3: Simulate the multi‑tenant multi‑tier multi‑domain denotes the number of tiers in a domain network. Ad‑
network with the aid of Alg. 1 and Alg. 2 ditionally, the cardinalities of the set of InPs or domain
4: Enter the outcome(s) of iteration in line 3 into networks, set of slice use‑cases, set of users belonging to
( ,:) a SP are represented as |ℐ|, | |, and | | respectively.

5: end for
6: Perform statistical analysis in (48) on 9. NUMERICAL RESULTS
7: Output result
We evaluate the performance of the M‑TTSD network
In a Monte Carlo simulation, a scenario (or an experi‑ resource allocation scheme via extensive Monte Carlo‑
ment) is run randomly over thousands of times. The out‑ based simulations in a MATLAB environment. We ex‑
comes are recorded for each iteration. The Monte Carlo amine a 2‑domain and 3‑domain network with InPs hav‑
result is based on the probability distribution of the sev‑ ing independent infrastructure and resources. Moreover,
eral iterations. the macro‑cells, picocells, femtocells, and clustered fem‑
© International Telecommunication Union, 2021 71