1. Other Markovian Systems
Queuing Models
Other Markovian Systems

2. 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.

3. 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.

4. 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.

5. 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 - / = = 4.504
W = L/  = 5.405/.4 = hrs.
WQ = Lq/  = 4.504/.4 = hrs.
There are no formulas for the pn’s!

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

7. 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 

8. 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

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

10. 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