# **Slotted ALOHA**
###### tags: `Project`

[TOC]

---
## **ALOHA**
:::spoiler ALOHA is the earliest and most basic wireless data communication protocol.
- It is a **Medium Access Control (MAC) protocol** for transmission of data via a shared network channel. Using this protocol, several data streams originating from multiple nodes are transferred through a multi-point transmission channel.
- There are two types of ALOHA protocols : Pure ALOHA and Slotted ALOHA.
:::

---
## **Pure ALOHA**
In pure ALOHA, the time of transmission is **continuous**.
<div style="text-align: center"><img src="https://i.imgur.com/BMUZNyE.jpg"/></div>

:::spoiler Working principle and characteristics
<div style="text-align: center"><img src="https://i.imgur.com/UvzfGgU.png"/></div>

- ++Working principle++ : Whenever a station has an available frame, it sends the frame to the channel immediately. If **an acknowledgement** is received within the finite period of time, it means the transmission is successful; otherwise it will be retransmitted.
- ++Retransmission strategy++ : Transmitting station uses a **Back Off Strategy**. If there is collision and the frame is destroyed, the sender waits for a random period of time before retransmitting it; if there is another conflict, wait for a random period of time until the retransmission is successful.
- ++Advantage++ : Simple and easy.
- ++Disadvantage++ : Conflict easily.
- ++Competitive system++ : A system where **multiple stations** share common channels in a way that may cause conflict. Trying to transmit at the same time, the frame and an error is detected by the receiving station and **no acknowledgement** is sent.
:::
:::spoiler Throughput Analysis
<div style="text-align: center"><img src="https://i.imgur.com/HkYRY72.png"/></div>

- Let ${\bf T}$ be the **frame transmission time** (assume constant) : the time required for a frame to be transmitted.
- Let ${\bf S}$ be the **throughput** : the average number of successful frame transmissions per frame transmission time ${\bf T}$.
- Let ${\bf G}$ be the **load** : the average number of attempted frame transmission attempts per frame transmission time ${\bf T}$.
- Let ${\bf P_{successful}}$ be the **probability** a frame transmission is successful.
$${\bf P_{successful}\{k \ transmissions \ in \ 2T \ seconds\}=\frac{(2G)^{k}}{k!}e^{-2G}=\frac{(λ2T)^{k}}{k!}e^{-λ2T}, \ k=0,1,2,...} \\ {\bf S=GP_{successful}}$$
- It is the **Poisson distribution** describing all transmissions over the channel where $${\bf λ=\frac{Transmission \ rate \ in \ frames}{Second}}$$
- This expression assumes an **infinite population** of infrequent stations.
:::
:::spoiler Vulnerable Period for Pure ALOHA
<div style="text-align: center"><img src="https://i.imgur.com/RnF5zse.png"/></div>

- Some frames (packets) being transmitted. The lower diagram shows the positioning of other frames that just avoid colliding with the original frame. The vunerable period can be seen to be of length ${\bf 2T}$.
- ${\bf P_{r}}$ is the probability of only one transmissions in the vulnerable period of length ${\bf 2T}$. (note that ${\bf λ=\frac{G}{T}}$)
$${\bf k=1, \ P_{successful}=P_{r}\{1 \ transmissions \ in \ 2T \ seconds\}=2Ge^{-2G}}$$
- Because $S$ is the throughput, so there's **no collision** (${\bf 0}$ frames are initiated in the vulnerable time period, ${\bf k=0}$)
$${\bf S=GP_{r}\{noCollision\}=GP_{r}\{0 \ transmissions \ in \ 2T \ seconds\}=Ge^{-2G}}$$
:::
:::spoiler Maximum Throughput of Pure ALOHA
<div style="text-align: center"><img src="https://i.imgur.com/drM78Mz.png"/></div>

- Throughput ${\bf S}$ has maximum value of ${\bf \frac{0.5}{e}≈0.184}$ at load ${\bf G=0.5}$, and then declines back toward 0.
- Two or more arrivals in a vulnerable period ($2T$) result in a collision.
- The system has two modes: for small ${\bf G}$ (i.e. ${\bf S≈G}$, there is no collision) and for large G (i.e. ${\bf G>>S}$, there are many backlogged users).
- Pure ALOHA system **cannot achieve throughput higher than** ${\bf 18.4\%}$.
:::

---
## **Slotted ALOHA**
In slotted ALOHA, the time of transmission is **discrete**.
<div style="text-align: center"><img src="https://i.imgur.com/6SqmCOK.jpg"/></div>

:::spoiler Working principle and characteristics
- ++Working principle++ : The channel time is divided into discrete time slots, and the slot length is the transmission time required for one frame. Any station can send only one frame at each slot. Also, the stations cannot transmit at any time whenever a frame is available, they should wait for the beginning of the next slot. The other process is the same as the Pure ALOHA protocol.
- ++Retransmission strategy++ : Same as Pure ALOHA. Transmitting station uses a **Back Off Strategy**. If there is collision and the frame is destroyed, the sender waits for a random period of time before retransmitting it; if there is another conflict, wait for a random period of time until the retransmission is successful.
- ++Cost++ : The whole network needs to be **synchronized**; a special site can be set up, and the site sends the clock signal.
- ++Three states of the STA++
- **Idle** (no ready frame to be transmitted)
- **Backlogged** (the frame transmitted in the previous slot was failed, and waiting for the next slot to retransmit)
- **Transmitting** (transmit new frame if the station was in idle state, or retransmit the failed frame if the station was in backlogged state)
:::
:::spoiler Throughput Analysis
<div style="text-align: center"><img src="https://i.imgur.com/GLozx9c.png"/></div>

- Let ${\bf T}$ be the **frame transmission time** (assume constant) : the time required for a frame to be transmitted.
- Let ${\bf S}$ be the **throughput** : the average number of successful frame transmissions per frame transmission time ${\bf T}$.
- Let ${\bf G}$ be the **load** : the average number of attempted frame transmission attempts per frame transmission time ${\bf T}$.
- Let ${\bf P_{successful}}$ be the **probability** a frame transmission is successful.
- Only frames that arrive during prior ${\bf T}$ seconds collide.
:::
:::spoiler Case of **infinite population** of stations : Vulnerable Period
<div style="text-align: center"><img src="https://i.imgur.com/y7DVWIV.png"/></div>

- The vulnerable period for Slotted ALOHA is reduced from ${\bf 2T}$ to ${\bf T}$ seconds. (note that ${\bf λ=\frac{G}{T}}$)
$${\bf P_{successful}\{k \ transmissions \ in \ T \ seconds\}=\frac{(G)^{k}}{k!}e^{-G}=\frac{(λT)^{k}}{k!}e^{-λT}, \ k=0,1,2,...}$$
- ${\bf P_{r}}$ is the probability of only one transmissions in the vulnerable period of length ${\bf T}$.
$${\bf k=1, \ P_{successful}=P_{r}\{1 \ transmissions \ in \ T \ seconds\}=Ge^{-G}} \\ {\bf S=GP_{r}\{noCollision\}=GP_{r}\{0 \ transmissions \ in \ 2T \ seconds\}=Ge^{-G}}$$
:::
:::spoiler Case of **infinite population** of stations : Maximum Throughput
<div style="text-align: center"><img src="https://i.imgur.com/h0J2X3a.png"/></div>

- Throughput ${\bf S}$ has maximum value of ${\bf \frac{1}{e}≈0.368}$ at load ${\bf G=1}$ in slotted ALOHA, which is an improvement over maximum throughput of ${\bf 18.4\%}$ in Pure ALOHA.
- Collision Probability Analysis
$${\bf P_{r}\{Collision\}=P_{r}\{> \ 1 \ transmissions \ in \ T \ seconds\} \\ =1-P_{r}\{0 \ transmissions \ in \ T \ seconds\}-P_{r}\{1 \ transmissions \ in \ T \ seconds\} \\ =1-P_{r}\{noCollision\}-P_{r}\{1 \ transmissions \ in \ T \ seconds\} \\ =1-e^{-G}-Ge^{-G}}$$
:::
:::spoiler Case of **finite population** of stations : ~~Vulnerable Period~~
- Revise the model for a finite population and determine under what conditions the infinite population model gives a reasonable approximation.
- Consider a network with ${\bf M}$ independent stations. The frame transmission from each station is modelled as a sequence of **independent Bernoulli Trials**.
- Let ${\bf P_{S_{i}}}$ be the probability of station ${\bf i}$ **successfully transmitting** a frame in any slot.
- Let ${\bf 1-P_{S_{i}}}$ be the probability of station ${\bf i}$ **not successfully transmitting** a frame in any slot.
- Let ${\bf P_{G_{i}}}$ be the probability of station ${\bf i}$ for **attempting transmissions** in any slot.
- Let ${\bf 1-P_{G_{i}}}$ be the probability of station ${\bf i}$ for **not attempting transmissions** in any slot.
- ${\bf i=1,...,M}$
- ${\bf S}$ is the system throughput (per slot), and ${\bf G}$ is the average offered load (per slot).
$${\cases{\bf S=\overset{M}{∑\limits_{i=1}}P_{S_{i}} \\ \bf G=\overset{M}{∑\limits_{i=1}}P_{G_{i}}}}$$
- The probability that station ${\bf j}$ has a successful transmission is the probability that station ${\bf j}$ is the **only station** to attempt a transmission in a particular slot.
$${\bf P_{S_{j}}=P_{G_{j}}\overset{M}{∏\limits_{i=1,i≠j}}{(1-P_{G_{i}})}} \\ { \bf Ex \ : \cases{\bf If \ j=1 \ succeeds \ : \ P_{S_{1}}=P_{G_{1}}(1-P_{G_{2}})(1-P_{G_{3}})...(1-P_{G_{M}}) \\ \bf If \ j=M \ succeeds \ : \ P_{S_{M}}=P_{G_{M}}(1-P_{G_{1}})(1-P_{G_{2}})...(1-P_{G_{M-1}})}}$$
- Assuming all stations share the load equally **(statistically identical)**.
$${\bf Identical \ : \ \cases{\bf P_{S_{j}}=P_{S_{1}}=P_{S_{2}}=...=P_{S_{M}}=P_{S}=\frac{Successfully \ transmission}{Every \ stations}=\frac{S}{M} \\ \bf P_{G_{j}}=P_{G_{1}}=P_{G_{2}}=...=P_{G_{M}}=P_{G}=\frac{Attempting \ transmission}{Every \ stations}=\frac{G}{M}}} \\ {\bf P_{S}=P_{G}(1-P_{G})^{M-1} \ → \ S=G(1-\frac{G}{M})^{M-1}}$$
- Note that ${\bf \lim\limits_{n→∞}(1+\frac{X}{n})^{n}=e^{X}}$. If we make ${\bf M→∞}$, we have the same result as the case of infinite population.
$${\bf S=\lim\limits_{M→∞}G(1-\frac{G}{M})^{M-1}=Ge^{-G}}$$
:::
:::spoiler Case of **finite population** of stations : Maximum Throughput
- For ${\bf M}$ stations the maximum throughput ${\bf S_{max}}$ is calculated by differentiating and equating to zero, then we have the same result as the case of infinite population.
$${\bf S(G)=G(1-\frac{G}{M})^{M-1} \ → \ \frac{dS(G)}{dG}=(1-\frac{G}{M})^{M-1}-\frac{G(M-1)(1-\frac{G}{M})^{M-2}}{M}} \\ {\bf Find \ maximum \ let \ S'=0 \ → \ (1-\frac{G}{M})^{M-1}-\frac{G(M-1)(1-\frac{G}{M})^{M-2}}{M}=0} \\ {\bf Divide \ the \ left \ and \ right \ by \ (1-\frac{G}{M})^{M-2} \ → \ (1-\frac{G}{M})-\frac{G(M-1)}{M}=0} \\ {\bf (1-\frac{G}{M})-G(\frac{M-1}{M})=1-G=0} \\ {\bf G=1} \ → \ {\bf S=S_{max}=G(1-\frac{G}{M})^{M-1}=(1-\frac{1}{M})^{M-1}}$$
- Collision Probability Analysis : the probability that at least two stations attempt transmission in a particular slot.
$${\bf P_{r}\{Collision\}=P_{r}\{> \ 1 \ attempt \ transmissions \ in \ particular \ slot\}=1-P_{r}\{0\}-P_{r}\{1\} \\ =1-\overset{M}{∏\limits_{i=1}}{(1-P_{G_{i}})}-P_{G_{j}}\overset{M}{∏\limits_{i=1,i≠j}}{(1-P_{G_{i}})}=1-(1-\frac{G}{M})^{M}-G(1-\frac{G}{M})^{M-1}}$$
- Conclusion
- During each time slot, each station will be transmitting the frame with probability $G_{i}=\frac{G}{M}$
- The successful transmission happens when there is only **one station transmitting the frame**
- The input from each station is modelled as a sequence of independent **Bernoulli Trials**
- There is no **backoff time**, the collided frame will be retransmitted again in the future time slot, based on the probability calculation of each station and **Bernoulli trial** (called **thinking state**)
:::

---
## **System Model of S-ALOHA**
- Expected Outcome
- Resource : [Page.33 & Page.35](https://www.slideshare.net/bhanutulya17/aloha-protocol-in-detail)
- While $G=1$, we have $S_{max}$
- We find $37\%$ empty, $37\%$ success, $26\%$ collision
<div style="text-align: center">
<img src="https://i.imgur.com/wJbJadG.png" width="300" />
<img src="https://i.imgur.com/cRKpAyh.png" width="300" />
</div>
:::spoiler Definition : Case of infinite-station
- **Simulation statistical parameters**
| Parameters | Definition | Value |
|:----------:|:------------------------------------:|:-----------------:|
| $M$ | Stations | $M→∞$ |
| $T$ | Total simulation time | $10^{4}$ |
| $G$ | Average attempted frame transmission | $0→10 \ by \ 0.1$ |
| $S$ | Number of simulation times | $10^{4}$ |
- **Simulation statistical variables**
| Variables | Definition | Range |
|:-------------------:|:------------------------------------------------------------------------------------:|:------------------:|
| $success\_frm[G]$ | The number of successful transmission | $0\sim∞ \ per \ G$ |
| $empty\_frm[G]$ | The number of idle transmission (no transmission) | $0\sim∞ \ per \ G$ |
| $collision\_frm[G]$ | The number of failed transmission | $0\sim∞ \ per \ G$ |
| $P[G]$ | Random value (of frames transmissions) related to Poisson Proba, the frame rate is G | $0\sim∞ \ per \ G$ |
| $t$ | Frame transmission time (each slot under each G) | $1\sim T$ |
| $s$ | The times of the simulation | $1\sim S$ |
- **Expected performance target**
| Target | Definition | Method |
|:-------------------------:|:--------------------------------------------------------------------------------:|:--------------------------:|
| $Success \ transmission$ | The number of successful frame transmission | $Ge^{-G}$ |
| $Average \ throughputs$ | The average number of successful frame transmissions per frame transmission time | $\frac{success\_frm}{T}$ |
| $Empty \ transmission$ | The number of idle frame transmission | $e^{-G}$ |
| $Average \ empties$ | The average number of idle frame transmissions per frame transmission time | $\frac{empty\_frm}{T}$ |
| $Collision \ probability$ | The probability of frame transmission occurs a collision | $1-e^{-G}-Ge^{-G}$ |
| $Average \ collisions$ | The average number of failed frame transmissions per frame transmission time | $\frac{collision\_frm}{T}$ |
- **Complement**
<div style="text-align: center">
<img src="https://i.imgur.com/8IChfGs.png" width="2000" />
</div>

- As the definition above, $P[G]$ is related to Poisson Probability. From the algorithm of Poisson random number (Gived by Donald Ervin Knuth), replace : $L=P_{r}$, $k=P[G]$, $λ=G$ and then we can get the random number.
- After the result of this algorithm, the average of $k$ is $λ$. It is as the mathematical theory of S-ALOHA, $G$ is an average of $P[G]$
- So, $P[G]$ means "the number of frames transmitted in a slot $t$".
:::
- **Flow chart : Case of infinite-station**

    st=>start: Start
    ops=>operation: s=1
    op=>operation: Success_frm = 0,
    Empty_frm = 0,
    Collision_frm = 0
    op0=>operation: G = 0
    op1=>operation: t = 1
    op2=>operation: P = number of frame
    transmitted based on G
    cond1=>condition: Is
    P=1?
    op3=>operation: Success_frm += 1
    cond2=>condition: Is
    P>1?
    op4=>operation: Empty_frm += 1
    op5=>operation: Collision_frm += 1
    cond3=>condition: Is
    t=T?
    op6=>operation: t += 1
    cond4=>condition: Is
    G=10?
    op7=>operation: S=Success_frm/T,
    E=Empty_frm/T,
    C=Collision_frm/T
    op8=>operation: G += 0.1
    cond5=>condition: Is
    s=S?
    op9=>operation: s += 1
    e=>end: End
    
    st->ops->op->op0->op1->op2->cond1
    cond1(yes)->op3->cond3
    cond1(no)->cond2
    cond2(yes)->op5->cond3
    cond2(no)->op4->cond3
    cond3(yes)->cond4->cond4
    cond3(no)->op6->op2
    cond4(yes)->op7->cond5
    cond4(no)->op8->op1
    cond5(yes)->e
    cond5(no)->op9->op

:::spoiler Definition : Case of finite-station
- **Simulation statistical parameters**
| Parameters | Definition | Value |
|:----------:|:----------------------------:|:-----------------:|
| $M$ | Stations | $10, 30, 50$ |
| $T$ | Total simulation time | $10^{4}$ |
| $λ$ | Attempted frame transmission | $0→10 \ by \ 0.1$ |
| $S$ | Number of simulation times | $10^{3}$ |
- **Simulation statistical variables**
| Variables | Definition | Range |
|:--------------------:|:-----------------------------------------------------------------:|:------------------:|
| $Total\_num\_frm[λ]$ | The number of attempted frame transmission in T | $0\sim∞ \ per \ λ$ |
| $success\_frm[λ]$ | The number of successful transmission | $0\sim∞ \ per \ λ$ |
| $empty\_frm[λ]$ | The number of idle transmission (no transmission) | $0\sim∞ \ per \ λ$ |
| $collision\_frm[λ]$ | The number of failed transmission | $0\sim∞ \ per \ λ$ |
| $Slot\_frm$ | The number of attempted frame transmission in a slot | $0\sim∞$ |
| $P \ (\frac{λ}{M})$ | Probability for attempting frame transmissions | $0\sim∞$ |
| $R$ | Generate a value meaning station has attempted frame transmission | $0\sim1$ |
| $m$ | The station id of a station range | $1\sim M$ |
| $t$ | Frame transmission time (each slot under each λ) | $1\sim T$ |
| $s$ | The times of the simulation | $1\sim S$ |
- **Expected performance target**
| Target | Definition | Method |
|:-------------------------:|:--------------------------------------------------------------------------------:|:----------------------------------------------:|
| $Average \ attempting$ | The average number of attempted frame transmission per frame transmission time | $\frac{Total\_num\_frm}{T}$ |
| $Success \ Transmission$ | The number of successful frame transmission | $G(1-\frac{G}{M})^{M-1}$ |
| $Average \ throughputs$ | The average number of successful frame transmissions per frame transmission time | $\frac{success\_frm}{T}$ |
| $Empty \ transmission$ | The number of idle frame transmission | $(1-\frac{G}{M})^{M}$ |
| $Average \ empties$ | The average number of idle frame transmissions per frame transmission time | $\frac{empty\_frm}{T}$ |
| $Collision \ probability$ | The probability of frame transmission occurs a collision | $1-(1-\frac{G}{M})^{M}-G(1-\frac{G}{M})^{M-1}$ |
| $Average \ collisions$ | The average number of failed frame transmissions per frame transmission time | $\frac{collision\_frm}{T}$ |
- **Complement**
<div style="text-align: center">
<img src="https://i.imgur.com/8IChfGs.png" width="2000" />
</div>

- As the definition above, $Slot\_frm$ is related to Poisson Probability. From the algorithm of Poisson random number (Gived by Donald Ervin Knuth), replace : $L=P \ (\frac{λ}{M})$, $k=Slot\_frm$, $λ=λ$, $u=R$ and then we can get the random number.
- After the result of this algorithm, the average of $k$ is $λ$. It is as the mathematical theory of S-ALOHA, $λ$ is an average of $Slot\_frm$
- So, $Slot\_frm$ means "the number of frames transmitted in a slot $t$".
:::
- **Flow Chart : Case of finite-station**

   st=>start: Start
   ops=>operation: s = 1
   op1=>operation: Total_num_frm = 0,
   success_frm = 0,
   Empty_frm = 0,
   collision_frm = 0,
   M = Number of STA
   op2=>operation: λ = 0
   op3=>operation: P = λ/M
   op4=>operation: t = 1
   op5=>operation: Slot_frm = 0
   op6=>operation: m = 1
   cond1=>condition: Is
   R<P?
   op7=>operation: Slot_frm += 1,
   Total_num_frm += 1
   op8=>operation: Do nothing
   cond2=>condition: Is
   m=M?
   op9=>operation: m += 1
   cond3=>condition: Is
   Slot_frm==1?
   op10=>operation: success_frm += 1
   cond4=>condition: Is
   Slot_frm>1?
   op11=>operation: collision_frm += 1
   op12=>operation: Empty_frm += 1
   cond5=>condition: Is
   t=T?
   op13=>operation: t += 1
   cond6=>condition: Is
   λ=10?
   op14=>operation: λ += 0.1
   op15=>operation: G = Total_num_frm/T,
   S = Success_frm/T,
   E = Empty_frm/T,
   C = Collision_frm/T
   cond7=>condition: Is
   s=S?
   op16=>operation: s += 1
   e=>end: End
   
   st->ops->op1->op2->op3->op4->op5->op6->cond1
   cond1(yes)->op7->cond2
   cond1(no)->op8->cond2
   cond2(yes)->cond3
   cond2(no)->op9->cond1
   cond3(yes)->op10->cond5
   cond3(no)->cond4
   cond4(yes)->op11->cond5
   cond4(no)->op12->cond5
   cond5(yes)->cond6
   cond5(no)->op13->op5
   cond6(yes)->op15->cond7
   cond6(no)->op14->op3
   cond7(yes)->e
   cond7(no)->op16->op1

---
## **Results : S-ALOHA Infinite-STA**
- **Source code of the simulation implementation**
- https://hackmd.io/@8KbRc796SnuYA2Dvsvk_BA/r10vmIpGu
- **Throughput S versus Load G for Slotted ALOHA :**
- When ${\bf G=1.000000}$, ${\bf S\cong0.367895}$
<div style="text-align: center">
<img src="https://i.imgur.com/4AlAhGf.jpg" width="500" />
<img src="https://i.imgur.com/gck2e1q.jpg" width="500" />
</div>
- **Empty E versus Load G for Slotted ALOHA :**
- When ${\bf G=1.000000}$, ${\bf E\cong0.367879}$
<div style="text-align: center">
<img src="https://i.imgur.com/ECOwRWO.jpg" width="500" />
<img src="https://i.imgur.com/NRkRFJo.png" width="500" />
</div>
- **Collision C versus Load G for Slotted ALOHA :**
- When ${\bf G=1.000000}$, ${\bf C\cong0.264241}$
<div style="text-align: center">
<img src="https://i.imgur.com/SSIU3TF.jpg" width="500" />
<img src="https://i.imgur.com/7qGTMGs.jpg" width="500" />
</div>
- **Conclusion :**
- While ${\bf G=1}$, we find $${\bf S+E+C\cong36.7895\%+36.7879\%+26.4241\%\cong100.0015\%\cong100\%}$$

---
## **Results : S-ALOHA Finite-STA**
- **Source code of the simulation implementation**
- https://hackmd.io/@8KbRc796SnuYA2Dvsvk_BA/r10vmIpGu
- **Throughput S versus Load G for Slotted ALOHA :**
- When ${\bf M=10}$, ${\bf G=1}$, ${\bf S\cong0.387420}$
- When ${\bf M=30}$, ${\bf G=1}$, ${\bf S\cong0.374133}$
- When ${\bf M=50}$, ${\bf G=1}$, ${\bf S\cong0.371602}$
<div style="text-align: center">
<img src="https://i.imgur.com/GOzbbZU.jpg" width="500" />
<img src="https://i.imgur.com/orDNeiK.jpg" width="500" />
</div>
- **Empty E versus Load G for Slotted ALOHA :**
- When ${\bf M=10}$, ${\bf G=1}$, ${\bf E\cong0.348678}$
- When ${\bf M=30}$, ${\bf G=1}$, ${\bf E\cong0.361662}$
- When ${\bf M=50}$, ${\bf G=1}$, ${\bf E\cong0.364170}$
<div style="text-align: center">
<img src="https://i.imgur.com/L5k7TAt.jpg" width="500" />
<img src="https://i.imgur.com/YAR4MOa.jpg" width="500" />
</div>
- **Collision C versus Load G for Slotted ALOHA :**
- When ${\bf M=10}$, ${\bf G=1}$, ${\bf C\cong0.263982}$
- When ${\bf M=30}$, ${\bf G=1}$, ${\bf C\cong0.264420}$
- When ${\bf M=50}$, ${\bf G=1}$, ${\bf C\cong0.264293}$
<div style="text-align: center">
<img src="https://i.imgur.com/QL4qxGB.jpg" width="500" />
<img src="https://i.imgur.com/piQe7Nk.jpg" width="500" />
</div>
- **Conclusion :**
- In ${\bf M=10}$, while ${\bf G\cong1}$, we find $${\bf S+E+C\cong38.7420\%+34.8678\%+26.3982\%\cong100.0080\%\cong100\%}$$
- In ${\bf M=30}$, while ${\bf G\cong1}$, we find $${\bf S+E+C\cong37.4133\%+36.1662\%+26.4420\%\cong100.0215\%\cong100\%}$$
- In ${\bf M=50}$, while ${\bf G\cong1}$, we find $${\bf S+E+C\cong37.1602\%+36.4170\%+26.4293\%\cong100.0065\%\cong100\%}$$
- If the station ${\bf M}$ is smaller, the throughput ${\bf S}$ will be larger, but it will decay to zero faster.
- If the station ${\bf M}$ is smaller, the empty ${\bf E}$ will be smaller, and it will decay to zero faster.
- If the station ${\bf M}$ is smaller, the collision ${\bf C}$ will approach 1 faster.

---
## **Reference**
- Jessica : ALOHA Background
- https://hackmd.io/@j-chen/HJzElTthw
- Bariq : Slotted ALOHA
- https://hackmd.io/@bariqsufif/slotted
- Poisson distribution
- https://zh.wikipedia.org/wiki/%E5%8D%9C%E7%93%A6%E6%9D%BE%E5%88%86%E5%B8%83
- Bernoulli Trials
- https://zh.wikipedia.org/wiki/%E4%BC%AF%E5%8A%AA%E5%88%A9%E8%A9%A6%E9%A9%97
- Others
- ALOHA
- https://baike.baidu.com/item/ALOHA/3695024
- http://bbcr.uwaterloo.ca/~lcai/ece418/5-2.pdf
- http://www.cse.cuhk.edu.hk/~cslui/CSC6480/wireless.pdf
- http://www.invocom.et.put.poznan.pl/~invocom/C/P1-4/p1-4_en/p1-4_3_1.htm
- Pure ALOHA
- https://www.tutorialspoint.com/pure-aloha
- S-ALOHA
- https://wangc86.github.io/csc0056_fall2019/lecture11_annotated2.pdf
- https://www.slideshare.net/bhanutulya17/aloha-protocol-in-detail
- Difference Between Pure ALOHA and S-ALOHA
- https://techdifferences.com/difference-between-pure-aloha-and-slotted-aloha.html
- https://www.tutorialspoint.com/differences-between-pure-aloha-and-slotted-aloha
- Simulation
- https://faculty.kfupm.edu.sa/COE/ashraf/RichFilesTeaching/COE032_543/Projects/Esa_present.ppt
- https://snipplr.com/view/80308/slotted-aloha-system