Simulation of M/M/1 queue with and without control of queue length.

mm1_queue.R focuses on generating data for further analysis.

various/fifo_example.c is a stand-alone program testing the data structure.

various/mm1_example.c is a stand-alone program testing the behaviour of the queue.

various/mm1_queue.ino is an Arduino implementation with heavy use of pointers. The queue is visualized using a LED dot matrix.

various/lora_03.ino is another Arduino implementation, using the queue as part of a data transmission project. In this version global variables are referenced directly to avoid the need for returning pointers to dynamically allocated memory.

Wikipedia:

In queueing theory, a discipline within the mathematical theory of probability, an M/M/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times have an exponential distribution. The model name is written in Kendall's notation. The model is the most elementary of queueing models and an attractive object of study as closed-form expressions can be obtained for many metrics of interest in this model.

General approach, following Bertsekas and Tsitsiklis (2008):

• Arrivals: Use a Bernoulli process as a discrete approximation of the Poisson process.
• Interpret a Bernoulli process as a sequence of independent Bernoulli random variables with probability p_1 of success at any given trial, a trial being, e.g., the flip of an unfair coin.
• Service times: Use a geometric distribution as a discrete approximation of the exponential distribution.
• Interpret a geometric random variable in terms of repeated independent trials with probability p_2 of success until the first success (again, a trial being, e.g., the flip of an unfair coin) .
• Discrete time: Loop with one iteration being one time step.
• In each iteration, use random number generator to generate arrivals with probability p_1 and departures with probability p_2. (Either one or zero arrivals per time step, either one or zero departures per time step).
• Keep track of queue length.
• Departures are only possible when queue length > 0 (after arrivals).
• Simple control mechanism: truncate every 10 steps to limit = x elements in queue.

The queue data structure (only in C programs) is implemented using an array, following Cormen, Leiserson, Rivest and Stein (2009), p. 234:

• Queue has a head and a tail.
• Enqueueing: inserting a new element at the tail of the queue.
• Dequeueing: removing the element at the head of the queue.
• Queue implemented with an array of size n (here: initialized with NULLs).
• Queued elements placed in a sequence of consecutive array slots.
• Enqueueing adds an element at the next slot to the right.
• Dequeueing removes the leftmost element (here: set to NULL, return value).
• Sequence of consecutive array slots "wraps around": slot 0 treated as the element following the last slot of the array.
• For simplicity, this description mostly refers to the case without wrapping around - the queue then consists of a sequence of consecutive array slots somewhere in the middle of the array (head leftmost element, tail empty slot to the right of the rightmost element).
• Overflow: tail reaches head, all array slots occupied. (To be avoided by sizing array appropriately. In case of overflow program should be terminated.)
• Returning NULL and no change of queue state in case of empty queue, thus avoiding underflow.
• Variable head used for index of element that will be dequeued next. Special cases:
  • Initial state: queue empty, head points to slot 0.
  • The last remaining element has been removed: head points to slot of element removed last.
• Variable tail used for index of empty slot to the right of the element that was enqueued last. Special cases:
  • Initial state: queue empty, tail points to slot 0.