Queuing Theory Non-Markov Systems

Motivation What happens if the system is not markovian; that is, we do not have exponential inter-arrival times and/or exponential service times. Three possible approaches Simulate the system with appropriate distributions Use other analytical approaches that approximate solutions through bounding techniques Ignore the underlying assumptions and approximate as an M/M/s/K model anyway

Motivation What happens if the system is not markovian; that is, we do not have exponential inter-arrival times and/or exponential service times. Three possible approaches Simulate the system with appropriate distributions Use other analytical approaches that approximate solutions through bounding techniques Ignore the underlying assumptions and approximate as an M/M/s/K model anyway

Pollaczek-Khintchine Formulation M/G/1 system Exponential inter-arrival times General service distribution with mean and standard deviation

P-K Formula

Suppose Service is Exponential If service is actually exponential, then Which is the formula for the M/M/1 model

Suppose Service is Exponential Further, if service is actually exponential, then Which again is the formula for the M/M/1 model

Non-Poisson System Approximation Suppose we have a general inter-arrival time and a general service distribution

Non-Poisson System Approximation define Then,

Non-Poisson System Approximation define Then, Note these are formulas for M/M/1 queue

Non-Poisson Approximation Suppose we do have exponential inter-arrival and exponential service times. Then,

Non-Poisson Approximation

Non-Poisson Approximation Again, if we have exponential inter-arrival and exponential service times, then and

M/D/1 Queue With the M/D/1 queue, we have exponential inter-arrival but deterministic service times Then,

M/D/1 Queue

M/D/1 Queue This is half the queue length and half the queue wait time as that of an M/M/1 queue. Why does this make sense?

M/D/1 Queue Because there is no variability in the service time. This is half the queue length and half the queue wait time as that of an M/M/1 queue. Why does this make sense? Because there is no variability in the service time.

Non-Poisson System Approximation Multiple Servers For a general inter-arrival time, general service distribution, and multiple servers