---
tags: EON, HSNL, NCKU
---

# Routing, Spectrum and Modulation Level Assignment, and Scheduling in Survivable Elastic Optical Networks Supporting Multi-Class Traffic
contributed by <`RusselCK` >

###### tags: `RusselCK`

- [ ] [Routing, Spectrum and Modulation Level Assignment, and Scheduling in Survivable Elastic Optical Networks Supporting Multi-Class Traffic](https://ieeexplore.ieee.org/document/8485690)

環境假設
* survivability of an EON with multiclass traffic
* each traffic type needs specific **protection mechanism** and **scheduling strategy**
* a connection request may demand for dedicated path protection

解法
* **Formulating an integer linear programming (ILP)** to serve multi-class traffic in EONs by considering various **survivability** and **scheduling** requirements.
* a number of **heuristic algorithms** to reduce the complexity of the ILP formulation in large scale networks
* two and three methods are considered for path selection and resource allocation, respectively.
* by comparing their performance with the ILP formulation

結論
* Our results reveal that by using load-aware path selection and first-fit frequency-slice and time-slot assignment strategies the best compromise between blocking probability and initial delay is obtained.

## I. Intoduction

#### The routing, modulation and spectrum assignment (RMSA) problem
* the routing and wavelength assignment (RWA) problem

#### Resource reservation schemes (the time domain behavior of traffic)

1. **immediate reservation (IR)**
* incoming requests must be served immediately upon their arrival time
2. **advance reservation (AR)**
* requests can tolerate a pre-determined delay and reserve resources in advance
* Considering the type of traffic (IR or AR) in the RMSA problem, increases its complexity

#### Alternative route (survivability policies)
1. provisioned or **reserved** before a failure occurs **(protection schemes)**
* [優] reduces the recovery time and improves the reliability of network
* [缺] degrades the spectrum efficiency and increases capital expenditure (CAPEX)
* backup resources may be assigned dedicatedly to a single connection or shared by multiple connections
* **Dedicated Path Protection (DPP)**
* **Shared Backup Paths Protection (SBPP)**
2. dynamically computed after the failure **(restoration schemes)**
* [優] higher spectrum efficiency and lower CAPEX
* [缺] lower reliability

## II. Problem Statement

### A. Network Model

* $G(V,~E)$: the network topology
* $V$: the set of nodes
* $E$: the set of links
* $\left| T\times F \right|$ resource elements (REs)
* $F$ frequency slices (FSs): the segmented spectrum of each fiber
* $T$ time slots (TSs): the divided time usage of FSs
* resource blocks (RBs): consisting of several adjacent REs

> $F=$ 4 ; $T=$ 8

> Rate and Supported Transmission Distance for Each Modulation Format in a FS With Bandwidth of 12.5 GHz

### B. Traffic Model

Static

* Assume that all the connection requests are **fixed and known**

Dynamic

* Assume that all the connection requests **arrive randomly**
and allocated resources to each request are released after the holding time of the request

* Two request arrival models:
1. Single
* at most one request arrives at each time slot
2. Batch
* a few number of requests arrive at each time slot

each request may be an IR or AR request
* IR request
* AR request
1. a specified starting time, and a specified duration (STSD)
* STSD-fixed: STSD with fixed starting time
> which means the request must start at a particular time in the future or it may be a flexible window, meaning that instead of a single starting time, a range of starting times are considered
2. a specified starting time but an unspecified duration (STUD)
3. an unspecified starting time but a specified duration (UTSD)

> the **time behavior of traffic**, i.e., its starting and holding times, must be included in resources allocation algorithms by considering time usages of spectrum slices.

Each **connection request**

$$
R(s, d, n, \delta, \zeta,\tau,\eta,\sigma)
$$

* $s$: source
* $d$: destination
* $n$: the number of required FSs in BPSK (the lowest modulation format)
* $\delta$: the holding time
* $\zeta$: the minimum starting time
* $\zeta=$ 1 for IR request
* $\tau$: the maximum ending time
* $\tau = \zeta + \delta -1$ for IR and STSD-fixed AR requests
* $\eta$: the survivability approach (protection or restoration)
$$
\eta =
\begin{cases}
1&\text{ if }~it~is~protection\\
0&\text{ if }~it~is~restoration
\end{cases}
$$
* $\sigma$ : the protection scheme (dedicated or shared)
$$
\sigma =
\begin{cases}
1&\text{ if }~it~is~dedicated\\
0&\text{ if }~it~is~shared
\end{cases}
$$

...
:::warning
TODO : example
:::

## III. The Proposed RMSA and Scheduling Algorithms to Support Multi-Class Traffic

### A. ILP Formulation

The optimization problem is formulated to minimize Bandwidth Block Probability (BBP)

**Objcetive**: maximize provisioned bandwidth demands
**Subjective**: continuity, contiguity and non-overlapping constraints

#### Notations

* $r=1,2,\ldots,R$ : Requests
* $t=1,2,\ldots,T$ : Time Slots
* $f=1,2,\ldots,F$ : Frequency Slices
* $p=1,2,\ldots,K$ : Candidate pairs of link-disjoint paths
* $t_{0}=\zeta,\zeta +1,\ldots,\tau -\delta$ : Candidate start time slots
* $f_{0}=1,2,\ldots,F-n_{p}$ : Candidate start frequency slices

### ILP problem

**Objective:**

$$
\begin{equation}
\begin{aligned}
\max\quad \left\lbrace
\sum \limits _{r}{{{S}_{r}}}\cdot n_{r}-\varepsilon \cdot \sum \limits _{r}{t_{r}^{0}}
\right\rbrace
\end{aligned}
\tag{1}
\end{equation}
$$

**Subject to:**

$$
\begin{align}
&\sum \limits _{p\in {{P}_{r}}}{\sum \limits _{t_{0},f_{0}}{x_{r,p}^{t_{0},f_{0}}=S_r}},\qquad \forall r;
\tag{2}\\
&\sum \limits _{p\in {{P}_{r}}}{\sum \limits _{t_{0},f_{0}}{y_{r,p}^{t_{0},f_{0}}={\eta _r}\times {S_r}}},\qquad \forall r;
\tag{3}\\
&{\eta _r}\times \sum \limits _{t_{0},f_{0}}{x_{r,p}^{t_{0},f_{0}}}= \sum \limits _{t_{0},f_{0}}{y_{r,p}^{t_{0},f_{0}}},\qquad \forall r,p\in {{P}_{r}};
\tag{4}\\
&{{\eta }_{r}}\times \sum \limits _{p\in {{P}_{r}}}{\sum \limits _{{{f}_{0}}}{x_{r,p}^{{{t}_{0}},{{f}_{0}}}}}=\sum \limits _{p\in {{P}_{r}}}{\sum \limits _{{{f}_{0}}}{y_{r,p}^{{{t}_{0}},{{f}_{0}}}}},\qquad \forall r,t_{0};
\tag{5}\\
&\sum \limits _{r}{\sum \limits _{p\in {{P}_{r}}}{\sum \limits _{t_{0},f_{0}}{\alpha _{r,p,t_{0},f_{0}}^{t,f}\,\delta _{r,p}^{e}\,x_{r,p}^{t_{0},f_{0}}}}}=a_{e}^{t,f},\qquad \forall e,t,f;
\tag{6}\\
&\sum \limits _{r}{\sum \limits _{p\in {{P}_{r}}}{\sum \limits _{{{t}_{0}},{{f}_{0}}}{\beta _{r,p,{{t}_{0}},{{f}_{0}}}^{t,f}\gamma _{r,p}^{e}((1-{{\sigma }_{r}})\cdot \delta _{r,p}^{g}+{{\sigma }_{r}})y_{r,p}^{{{t}_{0}},{{f}_{0}}}}}} \nonumber =b_{e,g}^{t,f},\qquad \forall e,g,t,f;\qquad
\tag{7}\\
&\sum \limits _{g}{b_{e,g}^{t,f}}\leq \theta \times c_{e}^{t,f},\quad \forall e,t,f;
\tag{8}\\
&c_{e}^{t,f} \leq \sum \limits _{g}{b_{e,g}^{t,f}},\quad \forall e,t,f;
\tag{9}\\
&a_{e}^{t,f}+c_{e}^{t,f}= v_{e}^{t,f},\qquad \forall e,t,f;
\tag{10}\\
&{{t}_{0}}\times \sum \limits _{p\in {{P}_{r}}}{\sum \limits _{{{f}_{0}}}{x_{r,p}^{{{t}_{0}},{{f}_{0}}}}}=t_{r}^{0}.
\tag{11}
\end{align}
$$

* (1): *minimize the BBP* by **maximizing the provisioned bandwidth demands**
* $S_r=1$, if $r$ is *provisioned*
* $n_r$ : the number of *FSs* of $r$
> The first term is *the provisioned bandwidth demands*
* $t_r^0$ : *the index of starting time slot* of $r$
> $\varepsilon$ : in order to be considered in selecting among solutions that have the same value for the first term.
> the second term is the summation of the starting times of the provisioned requests
:::warning
TODO : 理解 the second term
* (11) : assigns value to $t_r^0$ as the starting time slot index of $r$
:::
* (2): ensures that every *provisioned request* occupies a **unique RB**
* $x_{r,p}^{t_{0},f_{0}}=1$, if *the candidate RB* on primary $p$ (starts at $t_0$ and $f_0$ ) *be asigned to $r$*
* (3): same as (2) for *backup* paths
* (4): guarantees that allocated *primary and backup* **paths be chosen from the same candidate pair** of link-disjoint paths
* (5): assures that allocated **starting time slot** to *primary and backup* paths **be the same**
* (6): restricts link-joint *primary paths* **not to overlap** on a resource element on a particular link
* $\alpha _{r,p,t_{0},f_{0}}^{t,f}=1$ if *the candidate RB* of $r$ on primary $p$ (starts at $t_0$ and $f_0$ ) *uses $t$ and $f$*
* $\delta _{r,p}^{e}=1$, if *link $e$ belongs to primary $p$* of $r$
* $a_{e}^{t,f}=1$, if *RE $(t,f)$ is occupied on $e$* by a primary path
* (7): same as (6) for *backup* paths
* $b_{e,g}^{t,f}=1$, if RE $(t,f)$ is reserved on $e$ be used in the case of failure in link $g$
* (8)、(9):
* $c_{e}^{t,f}=1$, if RE $(t,f)$ is reserved on link $e$ to be used as backup
> It assures that if link $e$ be reserved as backup at RE $(t,f)$ in the case of failure in at least one of the network links, then $c^{t,f}_e=1$, otherwise as indicated in (9), $c^{t,f}_e=0$.
* (10): a RE on a particular link can not be allocated as a primary and backup, simultaneously.
* $v_{e}^{t,f}=1$, if RE $(t,f)$ is occupied or reserved as backup on link $e$

### Complexity Analysis


* Variables:
==$x_{r,p}^{t_{0},f_{0}}$==、$y_{r,p}^{t_{0},f_{0}}$、$a_{e}^{t,f}$、==$b_{e,g}^{t,f}$==、$c_{e}^{t,f}$、$v_{e}^{t,f}$、$S_r$、$t_r^0$
* Equality constraints:
(2)、(3)、==(4)==、==(5)==、(6)、==(7)==、(10)
* Imequality constraints:
==(8)==、(9)

### B. Heuristic Algorithm

1. two methods for **selecting** candidate pairs of disjoint **paths**
2. three strategies for **spectrum allocation and scheduling**

### 1) Candidate Pairs of Disjoint Paths Selection Methods

### (SP) K Shortest Pairs of disjoint paths
> Sorting benchmark : the **distance** of the routing paths

1. **sorted** all pairs of disjoint paths in an ascending order of the summation of the distances of their primary and backup paths.
2. **selects** first K pairs

### (LLP) K Least Loaded Pairs of disjoint path
> Sorting benchmark : parameter $B$

> select the routing paths which **balance the link bandwidth usage** while **utilizing smaller number of FSs**

$$
\begin{equation}
B(p)= \frac{ {{{MLBU}}_{p}^{w}}}{{m_{p}^{w}}}{\times} {{{HopCount}}_{p}^{w}}
+\frac{{{{MLBU}}_{p}^{b}}}{{{m}_{p}^{b}}}\times {{ {HopCount}}_{p}^{b}},
\tag{13}
\end{equation}
$$
> $b$ for backup

* $m_p^w$ : the *modulation formats* of the primary paths
* $HopCount_{p}^{w}$ : *the number of links* in the primary paths
* $MLBU_{p}$ : the link *bandwidth usage of the most loaded link* of path $p$
$$
\begin{align}
{{MLBU}_{p}^{w}}=&\underset{e}{\mathop {\max }}\,{{LBU}_{e}}\quad \forall e\in \text{working path }p,
\tag{14}\\
{{MLBU}_{p}^{b}}=&\underset{e}{\mathop {\max }}\,{{LBU}_{e}}\quad \forall e\in \text{backup path }p,
\tag{15}
\end{align}
$$
* $LBU_{e}$ : the *link bandwidth usage* of link $e$
> $e$ is adaptively updated when a request is provisioned

:::info
* $\frac{ {{{MLBU}}_{p}}}{{m_{p}}}$ try to **balance the link bandwidth usage**
* $HopCount_{p}$ try to **reduce resource utilization**
:::

---

* $LBU_{e}^{s}$ : the number of FSs of link $e$
* FSs are occupied by the primary paths or reserved for the backup paths of the provisioned requests at step $s$

as request $r$ is provisioned in step $s$, $LBU_{e}^{s}$ of all links in the primary path of request $r$ is updated

$$
\begin{equation}
LBU_{e}^{s+1}=LBU_{e}^{s}+{{n}_{r,p}^{w}}\quad
\forall e\in \text{working path }p,
\tag{16}
\end{equation}
$$

* ${{n}_{r,p}^{w}}$ : the number of required FSs for $r$ on primary path $p$

$$
\begin{equation}
n_{r,p}^{w}=\left\lceil {{n}_{r}}/{{m}_{p}^{w}} \right\rceil.
\tag{17}
\end{equation}
$$

> For DPP scheme:
>$$
\begin{equation}
LBU_{e}^{s+1}=LBU_{e}^{s}+{{n}_{r,p}^{b}}\quad \forall e\in \text{backup path }p,
\tag{18}
\end{equation}
>$$
>
> * ${{n}_{r,p}^{b}}$ : the number of required FSs for $r$ on backup path $p$
$$
\begin{equation}
n_{r,p}^{b}=\left\lceil {{n}_{r}}/{{m}_{p}^{b}} \right\rceil.
\tag{19}
\end{equation}
$$

> For SBPP scheme:
> $$
> \begin{equation}
> {LBU}_{e}^{s{+}1}={LBU}_{e}^{s}{+}{{n}_{r,p}^{b}}{-}\sum \limits_{f=f_{0}^{b}}^{f_{0}^{b}{+}{{n}_{b}}-1}{O_{e,f}^{s}}\;\;\; \forall e\in\text{backup path }p,
> \tag{20}
> \end{equation}
> $$
>
> * $O_{e,f}^{s}=1$, if FS $f$ is occupied or reserved on link $e$
> * $f_0^b$ : the starting FS index of the backup RB.

>When the protection scheme is SBPP, one RE may be reserved as backup resource for several requests with link-disjoint primary paths, but this RE must be considered once to calculate the link bandwidth usage.

### 2) Spectrum Allocation and Scheduling Strategies

1. **search** among available RBs in candidate pairs of disjoint paths computed for each request.
2. **select** the optimum primary and backup (if requested) RBs according to their objective.
* If there is no available RB, the request is removed.

### (LTW) Least Time to Wait

find **the least starting time slot** for request $r$ to be served as soon as possible.
* if among the available RBs with the same starting time slot, selects the one with **least starting frequency** index.

### (LSF) Least Starting Frequency slice index

find available RBs with **minimum starting frequency slice**
* if among those with the same starting frequency slice index, selects the one with **the least starting time slot**.

### (LTF) Least TF parameter

try to **minimize the starting frequency slice and time slot indices** at the same time

$$
\begin{equation}
TF= t_{0}+f_{0}^{w}+f_{0}^{b}
\tag{21}
\end{equation}
$$

* $f_0$ : the starting frequency slice indices
* $t_0$ : the index of starting time slot

### 3) Alogrithm


`#1~24` : **finds available RBs** as primary and also backup (if requested) for each request, along each candidate pair of link-disjoint paths.

* `#3` : **selects K candidate pairs of disjoint paths** among all pairs of link-disjoint paths for the source-destination pair of request r according to the **paths selection method**
* `#5` : selects the appropriate **modulation** level
* `#6` : calculates $n^w_{r,p}$ and $n^b_{r,p}$ using (17) and (19)
* `#9` : If no primary RB is found, it continues with next $t_0$
* `#10~15` : If a request **requires backup** resources, try to **find available backup RBs**
* `#16、17` : If at least one backup RB is found, sets **`flag = 1`**
* `#20` : Also if a request requires no backup resource, sets **`flag = 1`**

`#26` : If the **flag is set**, **selects the optimum primary** and backup (if requested) **RBs** and the associated path
* `#29` : otherwise the request is marked as blocked

## IV. Numerical Results

### A. Simulation Parameters

* $K=3$ : candidate pairs of disjoint paths

### 1) Static Scenario
:::success
* a number of requests are given
* algorithms are run one time to solve the RMSA and scheduling problems by considering all requests.
:::

* $a\%$ of all the given requests are **IR**
* $(1-a)\%$ of all the given requests are **AR**

- $b\%$ of each group ask for **protection mechanisms** (DPP or SBPP)
- $c\%$ request for **DPP**
- $(1-c)\%$ request for **SBPP**
- $(1-b)\%$ of each group ask for **restoration mechanisms**

w.l.o.g, we set $a=b=c=50\%$

#### small scale


* for each fiber link:
* 32 **FSs**
* **bandwidth** of 12.5 GHz
* **the look-ahead time** is set to 10 TSs
* for each request:
* **the source-destination pair** is selected randomly
* **the duration** follows a [negative exponential distribution](https://variation.com/wp-content/distribution_analyzer_help/hs122.htm) with average of 3 TSs
* **the transmission rate** is uniformly selected from the set $\lbrace \text{25}\,\text{Gbps}, \text{50}\,\text{Gbps}, \text{100} \,\text{Gbps}\rbrace$

#### large scale

> **NSFNet topology**
>- [ ] [Algorithms for the design of WDM translucent optical networks](https://)

* the network operates in the **C-band**
* **the bandwidth** of each fiber (4.475 THz) is divided into 358 FSs with 12.5 GHz granularity
* **the look-ahead time** is assumed to be 100 TSs
* for each request:
* **the source-destination pair** is selected randomly
* **the duration** follows a negative exponential distribution with average of 10 TSs
* **the transmission rate** is uniformly selected from the set $\lbrace \text{100}\,\text{Gbps}, \text{200}\,\text{Gbps}, \text{400}\, \text{Gbps}\rbrace$

### 2) Dynamic Scenario
> requests arrive dynamically

two arrival models:

#### 1. single arrival

* **the request arrival** at each TS follows a **Bernoulli distribution** with mean $α$
* if TSs are small and look-ahead time is large, the request arrival process can be approximated as a **Poisson process** with mean value of the look ahead time (in TSs) multiplied by α
* **the duration of each request** follows a negative exponential distribution with average of 20 TSs

#### 2. batch arrival

* **the number of arrived requests** follows a **Poisson distribution** with mean $β$
* **the duration of each request**follows a negative exponential distribution with average of 6 TSs

#### NSFNet topology

* a look-ahead time of 100 TSs
* a simulation duration of 300 TSs
> which is large enough to lead to small variance values for the results
* Transmission rate of each request is uniformly selected from the set $\lbrace \text{100}\,\text{Gbps}, \text{200}\,\text{Gbps}, \text{400}\, \text{Gbps}\rbrace$

### B. Evaluation Benchmark

### (BBP) Bandwidth Blocking Probability

BBP is defined as the aggregated bandwidth of all blocked requests over the total bandwidth of all incoming requests

$$
\begin{equation}
BBP=\frac{\sum \nolimits _{r\in \lbrace R-{{R}_{s}}\rbrace }{{{n}_{r}}}}{\sum \nolimits _{r\in R}{{{n}_{r}}}},
\tag{22}
\end{equation}
$$

* $R$ : the set of all arrived requests
* $R_s$ : the set of served requests.

### (AID) Average Initial Delay

Initial delay is defined as the difference between *the provisioned starting time* and *the minimum allowed starting time* of an AR request

$$
\begin{equation}
AID=\frac{\sum \nolimits _{r\in R_{s}^{a}}{(t_{r}^{0}-{{\zeta }_{r}})}}{\left| R_{s}^{a} \right|},
\tag{23}
\end{equation}
$$

* ${\zeta }_{r}$ : the minimum allowed starting time
* $t_{r}^{0}$ : the start time of request $r$
* $|R_{s}^{a}|$ : the number of served AR requests

### (ASE) Average Spectrum Efficiency

Average spectrum efficiency :=
*the total number of required FSs* in the case of lowest modulation format
/ the total number of assigned FSs
+
*the total number of required FSs for backup path* in the case of lowest modulation format
/ the total reserved FSs

$$
\begin{equation}
ASE=\frac{1}{2}\left(\frac{\sum \nolimits _{r\in {{R}_{s}}}{{{n}_{r}}}}{\sum \nolimits _{r\in {{R}_{s}}}{n_{r,p}^{w}}}+\frac{\sum \nolimits _{r\in {{R}_{s}}}{\eta \cdot {{n}_{r}}}}{\sum \nolimits _{r\in {{R}_{s}}}{\eta \cdot u_{r,p}^{b}}}\right)
\tag{24}
\end{equation}
$$

* $u^{b}_{r,p}$ : the number of idle FSs assigned to request $r$ on backup path $p$

$$
\begin{equation}
u_{r,p}^{b}={{\sigma }_{r}}\cdot n_{r,p}^{b}+(1-{{\sigma }_{r}})\cdot \left(n_{r,p}^{b}-\sum \limits _{f=f_{0}^{b}}^{f_{0}^{b}+{{n}_{b}}-1}{O_{e,f}^{s}}\right).
\tag{25}
\end{equation}
$$

### C. Simulation Results
:::warning
TODO:
1. 跑幾次?電腦設備?
1. why?
:::
### In static and small scale simulation.
> (a) the total provisioned bandwidth vs. the number of requests

> Note that results are obtained for the **SP** path selection method

for small request set size,
* all three heuristic algorithms serve the same amount of bandwidth

while with the growth of the number of requests
* the performance gap of the ILP formulation and the LTW algorithm is increased
* the LSF and LTF methods perform approximately the same as the ILP formulation.

---

> (b) the normalized AID vs. the request set size

:::warning
- [x] Why normalized AID ?

> the normalized AID = AID divided by the average of the permitted initial delay of the provisioned AR requests

* in the small scale simulations, as the request set size is not large enough to show accurate expectation values, the average permitted delay may differ in different cases
:::

* the LSF and LTF algorithms have a higher normalized AID with respect to the ILP formulation and LTW algorithm
* ILP formulation and LTW alternative have approximately the same normalized AID

:::info
This is due to the fact that ==the LSF and LTF algorithms **exploit the flexibility in the starting time of the AR requests** to increase the amount of served bandwidth==, which leads to lower BBP and higher AID
:::

---

> (c) The computational run times vs. the request set size

* the run time of the proposed ILP grows rapidly by increasing the request set size
* the three heuristic algorithms have the same run time

### In static and large scale simulations.

---

:::danger
In what follows, the results of large scale simulations are reported only for heuristic algorithms, because obtaining **the results of the ILP formulation is too time consuming in large scale simulations**, as well as the performance gap of the formulated ILP and the proposed heuristics is negligible.
:::

---

> (a) BBP vs the number of requests

* employing either LTF or LSF as the spectrum assignment and scheduling strategy mitigates the BBP
* by employing LLP as the path selection method, both the BBP and the AID are improved

:::success
the **LTF-LLP** method outperforms all the other alternatives
:::

---

> (b) AID vs the number of requests

* as LTF delays the AR requests less than LSF, LTF is preferred
* LTW delays the AR requests less than LTF and LSF

:::success
LTF > LSF > LTW
:::

* AID is not strictly increasing against the request set size

:::info
This is mainly because of the fact that ==when request set size grows, BBP increases and the number of provisioned AR requests decreases==, which may lead to less AID despite larger total number of AR requests
:::

---

> (c) ASE vs the number of requests

* by using LTF-LLP or LSF-LLP, spectrum efficiency is improved
* LTW-LLP has a higher spectrum efficiency than LTF-SP and LSF-SP

:::success
using **LLP** instead of SP increases the spectrum efficiency more than using LTF or LSF instead of LTW.
:::

* there is a **reverse relation between BBP and ASE** performances of all algorithms, unless the LTW-LLP algorithm.

:::info
This is because ==the LTW method tries to allocate preliminary time slots==, and as a result it increases the BBP of IR requests, so LTW-LLP shows worse performance in terms of BBP than LSF-SP and LSF-LLP
:::

* the algorithm which has more spectrum efficiency does not necessarily have less BBP

:::info
because besides trying to utilize FSs efficiently, it is also important to select TSs in an appropriate way which means ==using the flexibility of the starting times of the AR requests properly==.
:::

* ASE increases slightly with the growth of request set size

:::info
This is because in higher traffic load, more requests with disjoint-link primary paths may share the same resource elements on their common links, as a result, ==the sharing degree increases and this leads to a larger spectrum efficiency.==
:::

---

### In Dynamic with single arrival model.


> (a) BBP (b) AID (c) ASE vs $\alpha$ (the mean value of the Bernoulli distribution of arrival process)

> the same trend as the one observed in static and large scale
* ==**LTF-LLP** outperforms other algorithms==
* by using LTF-LLP or LSF-LLP, the BBP is reduced.
* as the LTF method delays the AR requests less than LSF, LTF-LLP is preferred
* the algorithms with larger ASE have less BBP, unless LTW-LLP
* LTW-LLP has higher ASE and BPP than LTF-LLP and LSP-LLP.

---

### In Dynamic with batch arrival model

> (a) BBP, (b) AID (c) ASE, vs. traffic load in dynamic with batch arrival model.
> * [Erlang](https://zh.wikipedia.org/wiki/%E5%9F%83%E6%9C%97%E5%96%AE%E4%BD%8D)
> - [ ] [What is an Erlang](https://www.electronics-notes.com/articles/connectivity/erlang/what-is-an-erlang-formula.php)
> - [ ] [What are Erlangs?](https://www.callcentrehelper.com/what-are-erlangs-154459.htm)

> the same trend as the one observed in static and large scale and in dynamic with single arrival model

## V. Conclusion

* the proposed algorithms have the same behavior in different simulation scenarios
* ==**LTF-LPP**== is the best algorithm in terms of BPP and ASE
* **LTW-SP** is the worst algorithm in terms of BPP and ASE
* **LTW-LLP** is the best in terms of AID

> we select the LTF-LPP algorithm as the most promising solution, which offers the best possible compromise between delay (AID) and blocking (BBP).