MATLAB Implementation of Queueing Theory with Exponential Distribution

Queueing theory is a mathematical discipline that studies waiting phenomena in service systems, while exponential distribution is commonly used to model the randomness of customer inter-arrival times or service times. Implementing exponential distribution simulation for queueing theory in MATLAB enables us to analyze system performance metrics such as average waiting time and queue length. To implement exponential distribution simulation for queueing theory, we first need to generate random numbers that follow exponential distribution. In MATLAB, we can use the built-in function `exprnd`, which generates exponential random variables based on a given mean parameter (the reciprocal of λ). For example, if the customer arrival rate λ=2 (meaning an average of 2 customers arrive per unit time), the inter-arrival time can be simulated using `exprnd(1/2)`. Basic queueing theory models typically include single-server or multi-server systems. Assuming we simulate an M/M/1 queue (single server, Poisson arrivals, exponential service times), we can follow these implementation steps: Generate arrival time sequence: Use the `exprnd` function to create customer inter-arrival times, then accumulate them to obtain actual arrival times for each customer. Generate service times: Similarly, use exponential distribution to simulate the processing time required for each customer. Calculate waiting times: Based on arrival times and service times, iterate through each customer to compute their waiting time and system sojourn time. Statistical performance metrics: Finally calculate key parameters such as average waiting time, queue length, and system utilization rate. For more complex models (like M/M/c multi-server systems), we can extend the above logic by adding server count determination and adjusting scheduling strategies (such as First-Come-First-Served). Through MATLAB simulation, we can visually observe how different parameters (such as arrival rate and service rate) affect system performance, thereby optimizing resource allocation or adjusting operational strategies. This approach is not only suitable for theoretical research but also applicable to practical scenarios such as efficiency analysis for call centers, supermarket checkout systems, and other service-oriented environments. Key implementation details include using vectorized operations for efficient computation, maintaining event timelines for accurate scheduling, and implementing proper queue management algorithms to handle customer flow dynamics. The `exprnd` function's statistical properties ensure realistic simulation of random processes, while MATLAB's plotting capabilities allow for visual analysis of system behavior over time.