Simple functions to calculate expected number of failures and workload in systems that can process a given number of subscribers in parallel, using a binomial distribution model to simulate subscriber behavior. Applicable e.g. to web servers & cellphone base stations.

workload_and_failures.py

from scipy.stats import binom

def workload(n_subscribers,n_capacity,p_call):
    """
    Calculates the estimated workload (in percent) of a system to which n_subscribers connect at any given time with probability *p_call* and that can process *n_capacity* clients in parallel.

    Arguments:

    n_subscribers    : The number of subscribers that might use the system at the given time.
    n_capacity       : The maximum number of subscribers that the system can process in parallel.
    p_call           : The probability that a subscriber will want to use the system at the given time.

    Return value:

    The expectation value of the workload of the system.
    """
    workload = 0
    n = 1.0
    while n < n_subscribers:
        if n < n_capacity:
            weight = n/float(n_capacity)
        else:
            weight = 1.0
        increment = weight*binom.pmf(n,n_subscribers,p_call)
        workload+=increment
        n+=1
    return workload

def failures(n_subscribers,n_capacity,p_call):
    """
    Calculates the expected number of processing failures in a system that can process *n_capacity* subscribers in parallel when *n_subscribers* will use it at a given moment with a probability *p_call*.

    n_subscribers    : The number of subscribers that might use the system at the given time.
    n_capacity       : The maximum number of subscribers that the system can process in parallel.
    p_call           : The probability that a subscriber will want to use the system at the given time.

    Return value:

    The expected number of processing failures (i.e. a subscriber wants to use the system but all slots are already filled).
    """
    failures = 0
    n = n_capacity+1.0
    while n < n_subscribers:
        increment = binom.pmf(n,n_subscribers,p_call)*(n-n_capacity)
        failures+=increment
        n+=1
    return failures