This post is based on the textbook Computer Networking: A Top-Down Approach (6th Edition) by James Kurose and Keith Ross.

1. Overview of Delay in Packet-Switched Networks

As a packet travels from source to destination, it experiences several types of delay at each node (router) along the path. The most important is **nodal delay**, which consists of four components.

Nodal delay (d_nodal) = d_proc + d_queue + d_trans + d_prop

d_proc : Processing Delay

d_queue : Queuing Delay

d_trans : Transmission Delay

d_prop : Propagation Delay

2. The Four Delay Components

2.1 Processing Delay

The time required to examine the packet header and determine where to direct the packet.

- Bit-level error checking of the packet header

- Forwarding table lookup

- Typically on the order of **microseconds (us)** or less

Packet arrives -> [Header check] -> [Forwarding decision] -> Enter queue

Processing delay (d_proc)

2.2 Queuing Delay

The time a packet waits in the output link's queue before transmission.

- Depends on the number of other packets waiting in the queue

- Ranges from **microseconds to milliseconds**

- **The most complex and unpredictable** of the four delays

Output queue:

[pkt5][pkt4][pkt3][pkt2][pkt1] --> Output link

(transmitting)

<-- Wait time for these packets = queuing delay -->

2.3 Transmission Delay

The time required to push all of a packet's bits onto the link.

Transmission delay = L / R

L: Packet length (bits)

R: Link transmission rate (bps)

**Example**: `L = 10,000 bits`, `R = 10 Mbps`

d_trans = 10,000 / 10,000,000 = 0.001 seconds = 1ms

> Transmission delay depends on **packet length** and **link transmission rate**, and is independent of the distance between routers.

2.4 Propagation Delay

The time it takes for a bit to physically propagate through the link.

Propagation delay = d / s

d: Physical distance between two routers (meters)

s: Propagation speed of the medium (approx. 2 * 10^8 m/s to 3 * 10^8 m/s)

**Example**: Distance between routers = 5,000 km, propagation speed = 2.5 x 10^8 m/s

d_prop = 5,000,000 / 250,000,000 = 0.02 seconds = 20ms

> Propagation delay depends on **distance** and is independent of packet size.

2.5 Transmission Delay vs Propagation Delay Analogy

Highway tollbooth analogy:

Car caravan = bits in a packet

Tollbooth = router

Highway = link

Transmission delay: Time for all cars to pass through the tollbooth

(number of cars / tollbooth processing speed)

Propagation delay: Time for one car to travel from one tollbooth to the next

(distance / car speed)

3. Queuing Delay and Packet Loss

3.1 Traffic Intensity

The extent of queuing delay can be assessed using **traffic intensity**.

Traffic intensity = L * a / R

L: Packet size (bits)

a: Average packet arrival rate (packets/sec)

R: Link transmission rate (bps)

Queuing Delay vs Traffic Intensity

Queuing

delay

^

| |

| | /

| | /

| | /

| | /

| __|_/

| ___/ |

|__/ |

+-----------+-------> Traffic intensity (La/R)

0 1

La/R -> 0 : Queuing delay nearly zero

La/R -> 1 : Queuing delay increases dramatically

La/R > 1 : Queue grows without bound (unstable system)

Key Rules

| Traffic Intensity | Queuing Delay Status |

| ------------------- | --------------------------------------------- |

| La/R close to 0 | Nearly zero |

| La/R close to 1 | Increases dramatically |

| La/R greater than 1 | Grows without bound (effectively packet loss) |

> Golden rule of system design: **Traffic intensity must not exceed 1**.

3.2 Packet Loss

In reality, queue (buffer) sizes are finite.

When a new packet arrives at a full buffer:

[pkt_n][...][pkt2][pkt1] --> Output link

^^^^^^^^^^^^^^^^^^^^^^^^

Buffer capacity = n (full)

pkt_new arrives -> Drop! (Packet loss)

- Lost packets may be **retransmitted** by the previous node or the source

- Or they may not be retransmitted at all (depends on the application)

- Loss rates increase significantly in congested networks

4. End-to-End Delay

Let us calculate the total delay from source to destination.

For N links (assuming no congestion):

d_end-to-end = N * (d_proc + d_trans + d_prop)

= N * (d_proc + L/R + d/s)

**Example**: 3 routers (3 links), `d_proc = 0.003ms`, `L = 1,500 bytes`, `R = 2 Mbps`, `d = 5,000 km`, `s = 2.5 * 10^8 m/s`

Delay per hop:

d_proc = 0.003 ms

d_trans = (1500 * 8) / 2,000,000 = 6 ms

d_prop = 5,000,000 / 250,000,000 = 20 ms

Per-hop delay = 0.003 + 6 + 20 = 26.003 ms

Total delay = 3 * 26.003 = 78.009 ms

4.1 Traceroute

The `traceroute` command (`tracert` on Windows) can measure the actual end-to-end path and per-hop delays.

traceroute www.example.com

Sample output:

1 192.168.1.1 1.234 ms 1.123 ms 1.345 ms

2 10.0.0.1 5.678 ms 5.432 ms 5.789 ms

3 72.14.215.85 15.234 ms 14.987 ms 15.123 ms

...

Each row represents one router (hop) with three RTT measurements.

4.2 Other Types of End-to-End Delays

Intentional Delays

- **Media packetization delay**: Time to collect voice data into packets in VoIP

- **Processing delay**: Virus scanning at email servers, etc.

5. Throughput

5.1 Definition

**Throughput** is the number of bits per unit time delivered from source to destination.

- **Instantaneous throughput**: Transmission rate at a specific point in time

- **Average throughput**: Average transmission rate over the entire transfer duration

Transferring a file of F bits takes T seconds:

Average throughput = F / T (bps)

5.2 Bottleneck Link

End-to-end throughput is determined by the **slowest link on the path**.

Server --Rs--> Router --Rc--> Client

Rs = Server-side link transmission rate

Rc = Client-side link transmission rate

Throughput = min(Rs, Rc)

Example 1: Server Side Is the Bottleneck

Server --2 Mbps--> Router --10 Mbps--> Client

Throughput = min(2, 10) = 2 Mbps

Bottleneck link: Server-side link

Example 2: Client Side Is the Bottleneck

Server --100 Mbps--> Router --1.5 Mbps--> Client

Throughput = min(100, 1.5) = 1.5 Mbps

Bottleneck link: Client-side link (access network)

5.3 Multiple Connections Sharing a Link

10 server-client pairs sharing one core link (R):

Server1 --Rs--+ +--Rc--> Client1

Server2 --Rs--+ +--Rc--> Client2

... +-- R (shared link) -+

Server10 --Rs--+ +--Rc--> Client10

Throughput per connection:

Throughput = min(Rs, Rc, R/10)

In the real Internet, core link capacity is very large, so the **access network** is usually the bottleneck.

Typical case:

Core link >> Access network transmission rate

-> Throughput = min(Rs, Rc)

-> Bottleneck is almost always the access network

6. Relationship Between Delay and Throughput

Delay and Throughput are independent performance metrics:

Delay: Time for a single packet to arrive

Throughput: Amount of data delivered per unit time

Pipe analogy:

+-----------------------------+

| Water (data) flowing |

| through a pipe |

+-----------------------------+

<-- pipe length = delay -->

pipe cross-section = throughput

7. Summary

| Delay Component | Depends On | Magnitude |

| ------------------ | --------------------------------- | ---------- |

| Processing delay | Router performance | us or less |

| Queuing delay | Traffic intensity | us to ms |

| Transmission delay | Packet size / link rate | us to ms |

| Propagation delay | Link distance / propagation speed | ms |

Key formulas:

d_nodal = d_proc + d_queue + d_trans + d_prop

Traffic intensity = La/R (must be less than 1 for stability)

Throughput = min(transmission rates of all links on the path)

8. Review Questions

- **Transmission delay**: The time to push all bits of a packet onto the link. Calculated as `L/R` and depends on packet size and link transmission rate.

- **Propagation delay**: The time for a bit to physically travel through the link. Calculated as `d/s` and depends on distance and propagation speed.

Tollbooth analogy: Transmission delay is the time for all cars to pass through the tollbooth; propagation delay is the time for one car to drive to the next tollbooth.

The average packet arrival rate exceeds the transmission capacity of the link, causing the queue to grow without bound. In reality, since buffers are finite, **massive packet loss** occurs. Therefore, networks must be designed so that traffic intensity does not exceed 1.

The slowest link on the path, the **bottleneck link**, determines end-to-end throughput. In the real Internet, the access network (home DSL, cable, etc.) is usually the bottleneck.