Other Markovian Systems Queuing Models Other Markovian Systems

MARKOVIAN SYSTEMS At least one of the arrival pattern or service time distribution is a Poisson process. There are many in which some of the conditions for M/M/k systems do not hold.

M/G/1 Systems M = Customers arrive according to a Poisson process at an average rate of  / hr. G = Service times have a general distribution with an average service time = 1/ hours and standard deviation of  hours (1/ and  in same units) 1 = one server Cannot get formulas for pn but can get performance measures using formulas.

Example -- Ted’s TV Repair Customers arrive according to a Poisson process once every 2.5 hours Repair times average 2.25 hours with a standard deviation of 45 minutes Ted is the only repairman: k= 1 THIS IS AN M/G/1 SYSTEM with:  = 1/2.5 = .4/hr. 1/ = 2.25 hours, so μ = 1/2.25 = .4444/hr.  = 45/60 = .75 hrs.

There are no formulas for the pn’s! Performance Measures The following are the hand calculations: P0 = 1-/ = 1-(.4/.4444) = .0991 L = (()2 + (/ )2)/(2(1-/ )) + / = ((.4)(.75)2 + (.4/.4444)2)/(2(.0991)) + (.4/.4444) = 5.405 LQ = L - / = 5.405 - .901 = 4.504 W = L/  = 5.405/.4 = 13.512 hrs. WQ = Lq/  = 4.504/.4 = 11.262 hrs. There are no formulas for the pn’s!

Input  (in customers/hr.)  (in customers/hr.)  (in hours) Performance Measures Select MG1 Worksheet

M/M/1 QUEUES WITH FINITE CALLING POPULATIONS (M/M/1//m) Maximum m school buses at repair facility, or m assigned customers to a salesman, etc. Both the arrival and service process are Poisson 1/ = average time between repeat visits for each of the m customers  = average number of arrivals of each customer per time period (day, week, mo. etc.) 1/ = average service time  = average service rate in same time units as 

Example -- Pacesetter Homes 4 projects Average 1 work stoppage every 20 days/project (Poisson Process) Average 2 days to resolve work stoppage dispute (Exponential Distribution) This is an M/M/1//m system with: m = 4 “arrival” rate of work stoppages,  = 1/20 = .05/day “service” rate,  = 1/2 = .5/day

Input , , m Performance Measures pn’s Select MM1 m Worksheet

Review An M/G/1 model is a single server model where the service time cannot be modeled as exponential, but has a mean time of 1/μ and standard deviation of service time, σ. Formulas exist for the steady state quantities for an M/G/1 system, but not for its, pn’s. An M/M/1//m is a single server system with a finite calling population of size m. Use of Templates MG1 MM1 m