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## Package umontreal.iro.lecuyer.contactcenters.ctmc

Provides call center simulators based on a quick uniformized continuous-time Markov chain model.

See:
Description

Interface Summary
AgentGroupSelector | Represents a policy selecting an agent group for an incoming contact, in a CTMC call center model.
CallCenterCTMC | Represents a continuous-time Markov chain (CTMC) modeling a call center with possibly multiple call types and agent groups.
CallCenterCTMCWithQueues | Extends the `CallCenterCTMC` interface for keeping track of the transition number for any queued contact.
CCEvent | Represents an event occurring during a transition of a CTMC representing a contact center with multiple contact types and agent groups.
CCEventFactory | Represents a factory for creating new call center events.
ProbInAWT | Represents an object that can compute information on the waiting time distribution conditional on the total number of transitions during a time horizon, the transition rate, and the number of transitions spent in queue by a particular contact.
TransitionListener | Represents a listener that can be notified when a transition occurs during a DTMC simulation of a call center.
WaitingQueueSelector | Represents a policy selecting a queued contact for an agent becoming free, in a CTMC call center model.

Class Summary
AbstractCallCenterCTMCSim | Base class for simulators of call centers using a continuous-time Markov chain.
AbstractCallCenterCTMCSimMP |
BasicCallCenterCTMCSim |
BasicCallCenterCTMCSimMP |
CallCenterCounters | Represents statistical counters computing sums for individual replications of a simulation of a call center using a discrete-time Markov chain.
CallCenterCTMC11 | CTMC model for a call center with a single call type and a single agent group.
CallCenterCTMC11WithQueues | Extension of the CTMC model for a single contact type and agent group, with information on contacts waiting in queue.
CallCenterCTMCKI | CTMC model of a call center with multiple call types and agent groups.
CallCenterCTMCKIWithQueues | Extends the CTMC model for multiple call types and agent groups with information on queued calls.
CallCenterCTMCQueues | Provides helper method used to maintain information on queued calls, for a CTMC model of a call center.
CallCenterStat | Regroups tallies collecting observations obtained from independent replications of a simulation using a CTMC in the case of an individual period.
CallCenterStatMP | Provides a merged view of several single-period `CallCenterStat` instances.
CircularIntArray | Represents a resizable circular array of integers.
EventWithSelection | Represents an event that can select the integers k = 0,…, K - 1 and p = 0,…, nk - 1 such that 0≤u - Wjnj - pWk < wk.
EventWithTest | Abstract event type with some helper methods to generate u1≤u < u2 randomly and uniformly, and to test that u < t for any value of t.
FalseTransitionEvent |
InitStateThresh | Used to initialize vectors of thresholds automatically.
IntMCallCenterCTMCSim |
ListGroupSelector | Represents an agent group selector using static lists for agent selection.
ListQueueSelector | Represents a waiting queue selector using static lists.
LookupEvent | Represents a call center event using random bits to select the index of a subinterval corresponding to an event.
PriorityGroupSelector |
PriorityQueueSelectorQS |
PriorityQueueSelectorWT |
ProbInAWTBinomial | Computes information on the conditional distribution of the waiting time, for a deterministic horizon.
ProbInAWTGamma | Computes information on the conditional distribution of the waiting time in the case of a random horizon.
QueueSizeThresh | Encapsulates thresholds on the queue size with the corresponding transition rates and geometric distributions for the number of successive self jumps preceding any generated transition.
RateChangeInfo | Represents information about a change in the arrival rate.
RateChangeTransitions | Provides methods to determine transitions at which arrivl rates change in order to have piecewise-constant arrival rates in the CTMC simulator.
SimpleGroupSelector |
SimpleQueueSelector |
StateThresh | Represents thresholds on the queue size, and the number of agents in each group.

Enum Summary
TransitionType | Represents the type of a transition performed by the `CallCenterCTMC.nextState(double)` method.

Exception Summary
CTMCCreationException |

## Package umontreal.iro.lecuyer.contactcenters.ctmc Description

Provides call center simulators based on a quick uniformized continuous-time Markov chain model. Staffing and scheduling optimization in large multiskill call centers is time-consuming, mainly because it requires lengthy simulations to evaluate performance measures and their sensitivity. Simplified models that provide tractable formulas are unrealistic in general. This package implements an intermediate solution, based on an approximate continuous-time Markov chain model of the call center. This model is more accurate than the commonly used approximations, and yet can be simulated faster than a more realistic simulation (based on non-exponential distributions and additional details). To speed up the simulation, the Markov chain is uniformized and only its discrete-time version is simulated. Performance measures such as the fraction of calls of each type answered within a given waiting time limit can be recovered from this simulation.

The CTMC model supports multiple call types and agent groups, but it assumes that calls arrive following a Poisson process, and service and patience times are exponential. For more information about the model, see [6].

The main class for the simulator is `BasicCallCenterCTMCSimMP`. It can simulate a call center with multiple periods by using one CTMC for each period. The `BasicCallCenterCTMCSim` class, on the other hand, provides a simulator for a single period, using a single CTMC. Both simulators first generate the total number of transitions from the Poisson distribution, followed by the transitions. However, the `IntMCallCenterCTMCSim` class implements a single-period simulator generating a large number of transitions and integrating over the number of transitions using Poisson probabilities when estimating performance measures.

All these simulators use CTMC models represented by the interface `IntMCallCenterCTMCSim`. The package provides two different implementations for this interface: one adapted for a single call type and agent group using linear search, and another one for multiple call types and agent groups based on indexed search.

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To submit a bug or ask questions, send an e-mail to Richard Simard.