Interpretation of network probability problem (transmission delay)

Question

According to the wiki, the transmission delay is the amount of time required to push all the packet's bits into the wire.

So I understand Probability $P (d_{tr,1} = k ∆_t)$ is that it takes $k$ times to transmit the entire a packet itself by device 1 in $P (d_{tr,1} = k ∆_t)$ chance. (Fail to send a packet $(k-1)$ times and Succeed to send it last one time $\geq k-1 + 1 = k$)

And from problem (a), I intuitively solve this question by inserting $1$ into $k$ because $∆_{t}$ means $k$ is $1$. But I can't understand its solution. I don't get how can it possible to divide this probability into two parts by using inequation such $P (t_{0} + d_{tr,1} ≤ t_{0} + ∆_t)$

I think $d_{tr,1}$ and $k∆_t$ have same meaning 'the transmission delay of the packet'. But this solution makes me confuse. I guess it tries to use cumulative geometrical distribution for solving problem and I cannot step more further unless understand the meaning of two symbols $d_{tr,1}$ and $k∆_t$.

What is the relationship between them? And how I can interpret this equation?

$P(t_{0} + d_{tr,1} ≤ t_{0} + ∆_t ) = P(d_{tr,1} ≤ ∆_t )$

Answer 1

• $t_0$ is the time transmission begins
• $d_{tr,1}$ is the transmission delay time for $1$ packet.
• $\Delta_t$ is the length of one time slot.
• $t_0+d_{tr,1}$ is the time that $1$ packet arrives; time of transmission plus transmission delay.
• $t_0+\Delta_t$ is the latest time we desire that packet to arrive; time of transmission plus one time slot.
• $\{t_0+d_{tr,1}\leqslant t_0+\Delta_{1}\}$ is the event that $1$ packet is delayed no later than one time slot after transmission begins. We seek the probability of this event.
• $\{d_{tr,1}\leqslant \Delta_{1}\}$ is the event that $1$ packet is delayed no longer than one timeslot. We can calculate the probability of this event.
  • These should be the same event as transmission delay is presumably independent of time of transmission.
• $\therefore~P(t_{0} + d_{tr,1} ≤ t_{0} + ∆_t ) ~=~ P(d_{tr,1} ≤ ∆_t )$

That is all.