How do you model complex systems with queuing networks?

1 What is queuing theory?

Queuing theory is the mathematical study of waiting lines, or queues, with the goal of understanding and predicting the behavior of systems that involve customers arriving at random times, requiring service from one or more servers, and leaving after being served. Examples of such systems include call centers, banks with multiple counters or ATMs, highways with toll booths or traffic lights, and hospitals that admit and treat patients in different wards or departments. Queuing theory can help answer questions such as how long a customer will have to wait in line, how many servers are needed to meet a certain service level, what is the optimal allocation of resources and capacity in the system, and how will the system react to changes in demand or service rates.

2 How to design a queuing network?

Designing a queuing network requires identifying and defining several elements, such as the arrival process which describes how customers arrive at the system, the service process which outlines how customers are served at each queue, and the departure process which outlines how customers leave the system. These elements include the distribution of interarrival times, the number of arrivals per time unit, the routing of customers to different queues, the distribution of service times, the number of servers, the service discipline (e.g., first-come first-served, priority, etc.), the probability of exiting or rejoining another queue, and the distribution of departure times.

3 How to analyze a queuing network?

To analyze a queuing network, you need to calculate or estimate some performance measures that indicate the efficiency and effectiveness of the system. These include the utilization, which measures how well the system uses its resources and capacity; the queue length, which measures the congestion and delay in the system; the waiting time, which measures the satisfaction and service quality of the system; and the throughput, which measures the productivity and output of the system. There are various methods and tools for analyzing queuing networks, such as analytical methods that use mathematical formulas to derive exact or approximate solutions for simple or standard queuing networks; simulation methods that use computer programs to generate scenarios of system behavior; and optimization methods that use mathematical models and algorithms to find optimal values for decision variables in order to maximize or minimize some objective function.

5 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?