Operations Research and Queuing Theory

Queuing Models

• Kendall's notation specifies the pattern of arrival, service time distribution, and number of channels in queuing models.

Game Theory

• The principle of dominance is used to eliminate strategies that would never be played by eliminating dominated strategies.
• A dominated strategy is one where a player can always do as well or better playing another strategy.

Arrival Distribution

• Arrivals can occur according to a known pattern or randomly.
• Random arrivals generally follow a Poisson distribution.

Service Time Distribution

• Service patterns can be constant or random.
• Constant service times are often machine-controlled.
• Random service times are often distributed according to a negative exponential probability distribution.

Markov Analysis

• Markov Analysis is a statistical technique used to model and analyze systems that transition from one state to another on a probabilistic basis.
• Key concepts of Markov Analysis include states, transition probabilities, transition matrix, and steady-state probabilities.
• Steps in Markov Analysis include identifying states, determining transition probabilities, constructing the transition matrix, and analyzing the transition matrix.

Markov Analysis Steps

• Identify the different states the system can be in.
• Calculate the probabilities of transitioning from one state to another.
• Create a matrix where each element represents the transition probability from one state to another.
• Study the system's behavior over time (transient analysis) and determine the long-term behavior of the system (steady-state analysis).

Equilibrium Conditions

• An equilibrium condition exists if state probabilities do not change after a large number of periods.
• At equilibrium, state probabilities for the next period equal the state probabilities for the current period.
• Equilibrium state probabilities can be computed by repeating Markov analysis for a large number of periods.

Game Theory Concepts

• Saddle point: a specific combination of strategies where neither player has an incentive to deviate.
• Value of the game: the average or expected outcome if the game is played repeatedly.
• Mixed strategy games: games where there is no saddle point, and players randomize their choices by playing each strategy with a certain probability.

Service Facility Design

• Service facility design may include parallel servers (e.g. post office or bank operation).
• Servers may also be arranged in series (e.g. jobs processed on successive machines) or networked (e.g. routers in a computer network).
• Types of service facilities include single-channel (one server), multi-channel (multiple servers), single-phase (customer receives service from just one server), and multi-phase (a customer has to go through more than one server).

Finite and Infinite Sources

• A finite source limits the customers arriving for service (e.g. machines requesting the service of a repairperson).
• An infinite source is forever abundant (e.g. calls arriving at a telephone exchange).