188. Queuing Theory
Queuing Theory represents the body of knowledge dealing with waiting lines.
Most queuing problems focus on determining the level of service that a company should provide.

189. Queuing Theory
Queuing Systems Configurations

190. Generation of Customers
Queuing Theory
Characteristics of a Queuing Process
Generation of Customers
Infinite vs. Finite calling population
Homogeneity of the calling population
Individual vs. Batch arrivals
Deterministic vs. Stochastic arrivals
Queuing of Customers
Single vs. Multiple servers
Finite vs. Infinite queues

191. Servicing the Customers
Queuing Theory
Characteristics of a Queuing Process
FIFO vs. LIFO disciplines
Priority rules
Servicing the Customers
Deterministic vs. Stochastic service time
Individual vs. Batch Processing

192. Generation of Customers
Queuing Theory
Characteristics of a Queuing Process
Generation of Customers
Poisson probability distribution
‘x’ represents the number of arrivals in a specific time period.
‘’ represents the ‘arrival rate’, that is, the average number of arrivals per time period.

193. Queuing Theory Arrival Rate
The time between arrivals is known as the interarrival time. If the number of arrivals in a given period follows a Poisson distribution, with mean , the interarrival times follow an exponential probability distribution with mean 1/
The exponential distribution exhibit the memoryless property. An arrival process is memoryless if the time until the next arrival occurs does not depend on how much time has elapsed since the last arrival.

194. Queuing Theory
Arrival Rate

195. Queuing Theory Service Rate
Queue time is the amount of time a customer spends waiting in line for service to begin.
Service time is the amount of time a customer spends at a service facility once the actual performance of service begins.
Service time is often model as an exponential random variable

196. Queuing Theory Service Rate
The service rate, denoted by , represents the average number of customers that can be served per time period. The average service time per customer is 1/ time periods.

197. Queuing Theory 1/2/3 Kendall Notation
The first characteristic identifies the nature of the arrival process using the following standard abbreviations:
M = Markovian interarrival times (following an exponential distribution)
D = Deterministic interarrival times (not random)

198. Queuing Theory Kendall Notation
The second characteristic identifies the nature of the service times using the following standard abbreviations:
M = Markovian service times
G = General service times (following a non-exponential distribution)
D = Deterministic service times (not random)
The third characteristic indicates the number of servers available.

199. Queuing Theory Operating Characteristics
U - Utilization factor, or the percentage of time that all servers are busy.
P0 - Probability that there are no units in the system.
Lq - Average number of units in line waiting for service
L - Average number of units in the system (in line and being served)
Wq - Average time a unit spends in line waiting for service
T - Actual time a unit spends in the queue
W - Average time a unit spends in the system (in line and being served)
Pw - Probability that an arriving unit has to wait for service
Pn - Probability of n units in the system

200. Queuing Theory The M/M/s Model
There are s servers in the system, where s is a positive integer
Arrivals follow a Poisson distribution and occur at an average rate of  per time period
Each server provides service at an average rate of  per time period, and actual service times follow an exponential distribution
Arrivals wait in a single FIFO queue and are serviced by the first available server
 < s

201. Queuing Theory
Formulas describing the M/M/s Model

202. Queuing Theory
Formulas describing the M/M/s Model

203. Queuing Theory
Q.xls

204. Queuing Theory
Case Problem (A) p. 140

205. Queuing Theory
Case Problem (cont.)

206. Queuing Theory
Case Problem (cont.)

207. Queuing Theory
Case Problem (cont.)

208. Queuing Theory
Case Problem (cont.)

209. Queuing Theory Finite Queue Model
Case Problem (cont.)

210. Queuing Theory Finite Queue Model
Case Problem (cont.)

211. Queuing Theory
Case Problem (cont.)

212. Queuing Theory
Case Problem (cont.)