Queueing Considerations: A System Of Two Servers Where Custo

Queueing Consider A System Of Two Servers Where Customers From Outs

Consider a system of two servers where customers from outside the system arrive at Server 1 at a Poisson rate of 4 and at Server 2 at a Poisson rate of 5. The service rates for Servers 1 and 2 are respectively 8 and 10. A customer, upon completion of service at Server 1, is equally likely to go to Server 2 or leave the system (i.e., P11 = 0, P12 = 0.5); whereas a departure from Server 2 will go 25% of the time to Server 1 and will depart the system otherwise (i.e., P21 = 0.25, P22 = 0). a. Draw a schematic of the system showing critical information; use Kendall’s notation to describe system characteristics. b. Determine the average number of customers and total expected time in the system.

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The queueing system described involves two interconnected servers with probabilistic customer movement both from outside arrivals and between servers. To analyze this system thoroughly, it is essential to visualize its structure, specify its characteristics with appropriate notation, and conduct calculations to determine key performance metrics such as the average number of customers and the expected total time within the system.

Schematic Representation and System Characteristics

The system can be depicted as two service nodes (Server 1 and Server 2), with external arrivals feeding into each node independently and some customers transitioning between the nodes before leaving the system. The crucial elements of this system include the Poisson arrival processes, exponential service times, and probabilistic transitions post-service.

External arrivals occur at Server 1 at a rate λ1 = 4 customers per unit time, and at Server 2 at λ2 = 5 customers per unit time. Service rates are μ1 = 8 at Server 1 and μ2 = 10 at Server 2, both assuming exponential service times characteristic of M/M/ systems.

Transition probabilities indicate that a customer finishing at Server 1 will move to Server 2 with probability 0.5 or leave the system, while a customer leaving Server 2 will either go back to Server 1 with probability 0.25 or exit the system entirely. These probabilistic flows imply a complex Markov process describing the state of the system.

Using Kendall’s notation, this system is classified as a two-node, Markovian network with routing and Poisson arrivals, often denoted as Jackson network. Since all arrivals are Poisson, and service times are exponential, with probabilistic routing, this network fits the criteria of a Jackson network, which allows for product-form solutions and simplifies analysis.

Analysis and Calculation of Performance Measures

To find the average number of customers and total expected time in the system, it is necessary first to determine the effective arrival rates to each server, considering the routing probabilities and external arrivals.

Step 1: Calculate Effective Arrival Rates

The effective arrival rates to servers (λ1' and λ2') account for external arrivals and routed customers from each server:

  • Effective arrival at Server 1: λ1' = external rate + customers routed from Server 2 to Server 1
  • Effective arrival at Server 2: λ2' = external rate + customers routed from Server 1 to Server 2

Let’s denote the external arrivals as λ1_ext = 4 and λ2_ext = 5. The flow equations, considering routing, are:

  • λ1' = λ1_ext + 0.25 × (average number of customers served at Server 2)
  • λ2' = λ2_ext + 0.5 × (average number of customers served at Server 1)

This forms a system of equations that can be solved iteratively or using linear algebra methods. For initial calculation, assume steady-state effective arrival rates are balanced so that:

λ1' = 4 + 0.25 × f2, and λ2' = 5 + 0.5 × f1, where f1 and f2 are the effective flows, proportional to arrival rates considering the routing probabilities.

In Jackson networks, the solution involves solving the flow balance equations:

λ1' = λ1_ext + P21 × λ2'


λ2' = λ2_ext + P12 × λ1'

Substituting known values:

λ1' = 4 + 0.25 × λ2'


λ2' = 5 + 0.5 × λ1'

Solving these equations simultaneously:

  • From the first: λ1' = 4 + 0.25λ2'
  • Substitute into the second: λ2' = 5 + 0.5(4 + 0.25λ2') = 5 + 2 + 0.125λ2'
  • Thus: λ2' - 0.125λ2' = 7, which simplifies to 0.875λ2' = 7
  • Therefore: λ2' = 7 / 0.875 ≈ 8
  • And λ1' = 4 + 0.25 × 8 = 4 + 2 = 6

Step 2: Calculate Mean Number of Customers at Each Server

In M/M/ systems, the expected number of customers (L) at each server is given by:

L = ρ / (1 - ρ), where ρ is the utilization, ρ = λ / μ.

Utilization at Server 1: ρ1 = λ1' / μ1 = 6 / 8 = 0.75

Utilization at Server 2: ρ2 = λ2' / μ2 = 8 / 10 = 0.8

Expected number of customers in each queue:

  • L1 = ρ1 / (1 - ρ1) = 0.75 / (1 - 0.75) = 3
  • L2 = ρ2 / (1 - ρ2) = 0.8 / (1 - 0.8) = 4

Total expected number of customers in the system:

L_total = L1 + L2 = 3 + 4 = 7

Step 3: Calculate Expected Time in the System

Using Little’s Law, W = L / λ, the average time a customer spends in each server is:

  • W1 = L1 / λ1' = 3 / 6 = 0.5 units of time
  • W2 = L2 / λ2' = 4 / 8 = 0.5 units of time

The total expected time spent by a customer in the system, accounting for the probabilistic routing, is approximately:

W_total = W1 + W2 = 1.0 units of time

This analysis provides a comprehensive view of the queueing system’s performance, demonstrating how probabilistic customer routing and server capacities influence the average customer load and wait times.

Conclusion

This study illustrates the application of Jackson network principles to a two-server queueing system with probabilistic routing and external arrivals. By systematically calculating effective arrival rates, utilizations, and applying Little’s Law, we derived meaningful performance metrics. Such models are vital in designing and optimizing complex service systems, ensuring efficiency and satisfactory customer experience.

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