Question

A router can process 2 million packets/sec. The load offered to it is \(1.5\) million packets/sec on average. If a route from source to destination contains 10 routers, how much time is spent being queued and serviced by the router?

Short answer

Total time spent: 20 seconds.

Step-by-step solution

Understanding the System Utilization

First, we need to determine the utilization of the router, which is the fraction of time the router is busy. Utilization can be calculated using the formula \( U = \frac{\text{offered load}}{\text{processing capacity}} \). In this exercise, the offered load is \(1.5\) million packets/sec, and the processing capacity is \(2\) million packets/sec. Therefore, the utilization \( U = \frac{1.5}{2} = 0.75 \). This means the router is busy 75% of the time.

Applying the Queueing Theory Formula

We'll use the formula for the average time spent in a single M/M/1 queue, which is \( T = \frac{1}{\mu - \lambda} \). In this formula, \( \lambda \) is the arrival rate, and \( \mu \) is the service rate. Here, \( \lambda = 1.5 \) million packets/sec and \( \mu = 2 \) million packets/sec. Plugging in these values, we find that \( T = \frac{1}{2 - 1.5} = \frac{1}{0.5} = 2 \) seconds spent at a single router.

Calculating Total Time for All Routers

Since there are 10 routers in the network path from source to destination, we need to multiply the average time spent at a single router by the number of routers. Thus, the total time spent being queued and serviced by all routers is \( 2 \times 10 = 20 \) seconds.

Key concepts

Router Utilization

Router utilization is a critical concept in queueing theory, especially when analyzing the performance and efficiency of a network. Utilization measures how much of the router's capacity is being used at any given time. In simpler terms, it tells us how busy a router is with processing incoming packets.

The utilization of a router is calculated by taking the ratio of the offered load to the router's processing capacity. For example, if a router can process 2 million packets per second and it receives 1.5 million packets per second on average, the utilization (denoted as \( U \)) is \[ U = \frac{1.5 ext{ million}}{2 ext{ million}} = 0.75. \]
This means the router is busy 75% of the time, indicating high utilization but also leaving some room (25%) for additional load without compromising performance.

High utilization can lead to increased waiting times, as packets may need to queue before being processed. However, extremely low utilization might mean resources are underutilized, which is inefficient from a capacity planning standpoint. Thus, striking a balance is important to maintain network efficiency without excessive delays.

Packet Processing

Packet processing is the core function of a router, where it receives and forwards packets to their appropriate destinations. Routers manage data packets by reading their headers, determining the best path forward, and ensuring data is delivered accurately and promptly.
Understanding packet processing involves knowing the router's processing capacity, which affects how many packets it can handle in a second. In the given exercise, we see that the processing capacity is 2 million packets per second. When incoming packets exceed this rate, queues start to form, leading to potential delays.

• Capacity: The maximum number of packets a router can process in a given period.
• Arrival Rate: The average number of packets arriving at the router. If this exceeds the capacity, packets will be queued.
• Queuing Delay: The waiting time experienced by packets before being processed when the router is highly utilized.
  
  

Efficient packet processing aims to minimize queuing delay by balancing the router's workload and optimizing packet paths through the network.

M/M/1 Queue Model

The M/M/1 Queue Model is a simplistic yet vital framework used in queueing theory to describe these types of systems. It's a mathematical model that helps us understand the behavior and performance of queues seen in routers.

"M/M/1" stands for:

• "M" for memoryless or Markovian arrivals: Packet arrivals follow a Poisson process, meaning each arrival is independent of the last.
• "M" for memoryless or Markovian service times: Service times (i.e., how long it takes to process a packet) are exponentially distributed.
• "1" denoting a single server: In this context, the "server" is the router's capacity to process packets.
  
  

The model simplifies the complexity of network processes by assuming random arrivals and services, making it easier to predict queuing behavior. In our exercise, to find the average time a packet spends in a queue, we use the formula \( T = \frac{1}{\mu - \lambda} \), where \( \lambda \) is the packets' arrival rate and \( \mu \) is the service rate. This formula gives us a tangible number for planning and optimizing routers' throughput, ensuring minimal disruptions and optimal performance.