Performance Evaluation of a Network Link (Queuing Theory in Networks)

To analyze network performance, it's essential to understand the nature of the data traffic:

Arrival Process:

Describes how packets arrive at a queue (e.g., a router's input or output buffer).
- Poisson Process (Random Arrivals): Often used as a simplifying assumption in queuing models. It implies that packet arrivals are independent and random, meaning the probability of an arrival in a short interval is proportional to the interval length, and previous arrivals don't affect future arrivals. This models relatively smooth, uniformly distributed traffic.
- Bursty Traffic: Real-world network traffic is frequently bursty, meaning packets arrive in short, intense bursts followed by periods of inactivity. This is more challenging for networks to handle efficiently than smooth traffic, as it can quickly overwhelm buffers and lead to congestion and loss.

Service Time Distribution:

Describes the time it takes for a packet to be processed or transmitted (the "service").
- For a network link, service time is typically determined by the packet length and the link's bandwidth (rate). (Service Time = Packet Length / Link Rate).
- If packet lengths vary, service times will also vary.

Traffic Intensity / Utilization (ρ):

This is a critical dimensionless parameter representing the ratio of the average arrival rate (λ) to the average service rate (μ) of a system.
- Formula: ρ = λ / μ
- It indicates the proportion of time a resource (e.g., a network link or a router port) is busy. If ρ approaches or exceeds 1, it signifies that the arrival rate is equal to or greater than the service rate, leading to rapidly growing queues, unbounded delays, and eventually significant packet loss. Network designs aim for ρ well below 1 to ensure stable operation.

When evaluating a network link or a router queue, several quantitative metrics are used:

• Average Number of Packets in the System (L): The average total count of packets present within the queuing system, including those waiting in the queue and those currently being transmitted (served).
• Average Number of Packets in the Queue (Lq): The average count of packets that are exclusively waiting in the buffer, not yet being transmitted.
• Average Waiting Time in the System (W): The average total time a packet spends from its moment of arrival until it successfully leaves the system (i.e., its time spent waiting in the queue plus its time spent being transmitted). This is often referred to as end-to-end delay when considering a series of links.
• Average Waiting Time in the Queue (Wq): The average time a packet spends solely waiting in the buffer before its transmission begins. This is the delay introduced by congestion at the queue.
• Packet Loss Probability: The probability that an arriving packet will be discarded (dropped) because the buffer (queue) at the router port is full and cannot accommodate any more incoming packets. This is a crucial measure of network congestion.
• Throughput: The actual rate at which packets or bits are successfully transmitted through the network link over a given period. It represents the effective data transfer rate, which is often less than the link's theoretical maximum bandwidth due to overhead, collisions, or congestion.

Kendall's notation is a standard shorthand used to classify and describe the fundamental characteristics of a queuing system. While detailed mathematical derivations are beyond this conceptual introduction, understanding the notation helps in identifying and discussing different types of network queues.

General Format: A / B / C / K / N

• A: Arrival Process Distribution: Describes the probability distribution of the inter-arrival times (the time intervals between consecutive packet arrivals).
• M (Markovian): Inter-arrival times follow an exponential distribution, implying random, independent arrivals.
• D (Deterministic): Inter-arrival times are fixed and constant.
• G (General): Inter-arrival times follow an arbitrary (general) probability distribution.
• B: Service Time Distribution: Describes the probability distribution of the time it takes to serve (transmit) a packet.
• M (Markovian): Service times follow an exponential distribution.
• D (Deterministic): Service times are fixed and constant.
• G (General): Service times follow an arbitrary (general) probability distribution.
• C: Number of Servers: Represents the number of parallel service channels available in the system. For a single router output link, this is typically 1.
• K (Optional): System Capacity (Buffer Size): The maximum number of packets that can be present in the entire queuing system (including those in the queue and those being served).
• N (Optional): Population Size: The total number of potential sources (customers) that can generate arrivals. If omitted, it implies an infinite population.

Common Model Example: M/M/1 Queue

This is a single-server queuing system where both the arrival process and the service time distribution are exponential (Markovian). The system has a single server (C=1) and is typically assumed to have infinite buffer capacity (K is omitted) and an infinite population (N is omitted). The M/M/1 model is a fundamental building block in network performance analysis. It's often used to conceptually model a single router output port with incoming packet traffic. It clearly demonstrates how performance metrics like average waiting time and queue length increase dramatically as the link utilization (traffic intensity ρ) approaches 1, highlighting the importance of managing network load.