Raik Stolletz

Approximation of the non-stationary M(t)/M(t)/c(t)-queue using stationary queueing models: The stationary backlog-carryover approach

This paper proposes a new approach for the time-dependent analysis of stochastic and non-stationary queueing systems. The analysis of a series of stationary queueing models leads to a new approximation of time-dependent performance measures. Based on a stationary backlog-carryover (SBC) approximation of the time-dependent expected utilization, different approximations of the time-dependent expected queue length and the number of customers in the system are discussed. Limiting results are given for the case of constant rates. The accuracy of the SBC approach is shown for non-stationary M(t)/M(t)/c(t) queueing systems with time-dependent and piecewise constant arrival rates. In numerical experiments we demonstrate the reliability of this approach and compare it with the (lagged) stationary independent period by period (SIPP) approach. In addition, the approximation is applied to temporarily overloaded systems that cannot be analyzed by the variants of the SIPP approach.

Performance Analysis and Optimization of Inbound Call Centers

1 Introduction.- 2 Characterization of Inbound Call Centers.- 2.1 What is an Inbound Call Center?.- 2.2 Performance Measures and Objective Functions.- 2.2.1 Technical Performance Measures.- 2.2.2 Economic Performance Measures.- 2.2.3 Objective Functions for Operational Planning.- 2.3 Operational Personnel Planning in Inbound Call Centers.- 2.3.1 Main Tasks of Operational Personnel Planning.- 2.3.2 Call Forecasting.- 2.3.3 Weekly Personnel Scheduling.- 2.3.3.1 Current Planning Process in Practice.- 2.3.3.2 Agent Requirements Planning.- 2.3.3.3 Shift Scheduling and Rostering.- 2.3.3.4 A Simultaneous Planning Approach.- 3 Classification of Queueing Models of Inbound Call Centers.- 3.1 Characteristics of Queueing Models of Call Centers.- 3.2 Classification by Customers and Agents.- 3.2.1 Arrival Process.- 3.2.2 Waiting Behavior.- 3.2.3 Distribution of Service Time.- 3.2.4 Homogeneity of Customers and Agents.- 3.3 Routing Decisions in Call Centers.- 3.3.1 Overview.- 3.3.2 Customer Selection.- 3.3.3 Agent Selection.- 3.4 Limitation of the Waiting Room.- 3.5 Review of the Literature.- 3.5.1 Overview.- 3.5.2 Homogeneous Customers and Homogeneous Agents.- 3.5.3 Heterogeneous Customers and Heterogeneous Agents.- 3.5.3.1 Introduction of the M-Design.- 3.5.3.2 The M-Design with Priority-Based Routing Policies.- 3.5.3.3 Special Cases of the M-Design with Priority-Based Routing Policies.- 4 Queueing Models of Call Centers with Homogeneous Customers and Homogeneous Agents.- 4.1 Common Features.- 4.2 The M /M/c and M /M/c/K Queueing Models with Patient Customers.- 4.2.1 Description and Derivation of Performance Measures for the M /M/c/K Queueing Model.- 4.2.2 Description and Derivation of Performance Measures for the M /M/c/00 Queueing Model.- 4.2.3 Numerical Results.- 4.2.3.1 Impact of the Number of Trunks.- 4.2.3.2 Economies of Scale.- 4.2.3.3 Impact of Talk Time.- 4.2.3.4 Optimal Number of Agents and Trunks in Large Call Centers.- 4.3 An M /M/c/K Queueing Model with Impatient Customers.- 4.3.1 Model Description and Derivation of Performance Measures.- 4.3.2 Numerical Results.- 4.3.2.1 Impact of Customer Impatience: Reneging 8.- 4.3.2.2 Impact of Customer Impatience: Balking 8.- 4.3.2.3 Impact of Customer Impatience: Dependencies between Reneging and Balking 8.- 4.3.2.4 Optimal Number of Agents and the Impact of the Number of Trunks 9.- 4.4 Management Implications of the Numerical Results.- 5 Queueing Model of a Call Center with two Classes of Customers and Skill-Based Routing.- 5.1 Description of the Queueing System.- 5.2 Description of the State Space.- 5.2.1 Representation of the States.- 5.2.2 Division and Size of the State Space.- 5.3 Steady-State Equations.- 5.3.1 Steady-State Equations for States with Waiting Aand B-Customers.- 5.3.2 Steady-State Equations for States with Waiting Aand without Waiting B-Customers.- 5.3.3 Steady-State Equations for States with Waiting Band without Waiting A-Customers.- 5.3.4 Steady-State Equations for States without Waiting Customers.- 5.4 Determination of Performance Measures.- 5.4.1 Solution of the Steady-State Equations.- 5.4.2 Derivation of Technical Performance Measures.- 5.4.3 Validation of the Derivation for Special Cases.- 5.5 Numerical Results.- 5.5.1 Impact of Flexible Agents: Effects of Priority-Based Customer Selection and Pooling.- 5.5.1.1 Impact of Priority-Based Customer Selection.- 5.5.1.2 Comparison of Call Centers with Completely Crosstrained or Completely Specialized Agents.- 5.5.1.3 Advantages of Adding Generalists Instead of Specialists.- 5.5.2 Impact of the Processing Times of Generalists.- 5.5.3 Impact of the Allocation of Trunks.- 5.5.4 Impact of the Allocation of a Fixed Number of Agents.- 5.5.4.1 Identically Distributed Processing Times for Specialists and Generalists.- 5.5.4.2 Different Processing Times for Generalists and Specialists.- 5.6 Management Implications of the Numerical Results.- 6 Conclusions and Suggestions for Further Research.- A Algorithms for Call Center Models with Homogeneous Customers and Agents.- A.1 Computations for the M /M/c/K Model with Patient Customers.- A.2 Computations for the M /M/ c/00 Model with Patient Customers.- A.3 Computations for the M /M/c/K Model with Impatient Customers.- A.3.1 Computation of Steady-State Probabilities.- A.3.2 Computation of the Waiting Time Distributions.- B Appendix for the Queueing Model of a Call Center with two Classes of Customers and Skill-Based Routing.- B.1 Derivation of the Number of States.- B.2 Derivation of the Remaining Steady-State Equations.- B.2.1 Steady-State Equations for States with Waiting Aand without Waiting B-Customers.- B.2.2 Steady-State Equations for States with Waiting Band without Waiting A-Customers.- B.2.3 Steady-State Equations for States without Waiting Customers.- B.3 Algorithms Used for Performance Analysis.- B.3.1 Description of Data Structures and Algorithms.- B.3.2 Behavior of the Algorithm.- Glossary of Notation.- List of Figures.- List of Tables.- References.

Fair optimization of fortnightly physician schedules with flexible shifts

This research addresses a shift scheduling problem in which physicians are assigned to demand periods. We develop a reduced set covering approach that requires shift templates to be generated for a single day and compare it to an implicit modeling technique where shift-building rules are implemented as constraints. Both techniques allow full flexibility in terms of different shift starting times and lengths as well as break placements. The objective is to minimize the paid out hours under the restrictions given by the labor agreement. Furthermore, we integrate physician preferences and fairness aspects into the scheduling model. Computational results show the efficiency of the reduced set covering formulation in comparison to the implicit modeling approach.

Aircraft landing problems with aircraft classes

This article focuses on the aircraft landing problem that is to assign landing times to aircraft approaching the airport under consideration. Each aircraft’s landing time must be in a time interval encompassing a target landing time. If the actual landing time deviates from the target landing time additional costs occur which depend on the amount of earliness and lateness, respectively. The objective is to minimize overall cost. We consider the set of aircraft being partitioned into aircraft classes such that two aircraft of the same class are equal with respect to wake turbulence. We develop algorithms to solve the corresponding problem. Analyzing the worst case run-time behavior, we show that our algorithms run in polynomial time for fairly general cases of the problem. Moreover, we present integer programming models. We show by means of a computational study how optimality properties can be used to increase efficiency of standard solvers.

Non-stationary delay analysis of runway systems

This paper proposes a new approach for the estimation of aircraft delays at airports given time-varying demand and time-dependent processing times. Based on a characterization of performance models for runway systems, the analysis addresses a runway system where arrivals and departures share a common runway. It is assumed that the requests for landings and take-offs can be modeled as independent Poisson processes and that the processing times are generally distributed with operation-dependent rates. The runway is operated according to the first-come-first-serve rule. A stationary backlog-carryover (SBC) approach is developed for the approximation of time-dependent performance measures for this dynamic queueing system with mixed operations. Numerical examples demonstrate that the SBC approximation is reliable for the analysis of runway queues. Because of its simplicity, the approach is numerically stable and fast.

Buffer allocation in stochastic flow lines via sample-based optimization with initial bounds

The allocation of buffer space in flow lines with stochastic processing times is an important decision, as buffer capacities influence the performance of these lines. The objective of this problem is to minimize the overall number of buffer spaces achieving at least one given goal production rate. We optimally solve this problem with a mixed-integer programming approach by sampling the effective processing times. To obtain robust results, large sample sizes are required. These incur large models and long computation times using standard solvers. This paper presents a Benders Decomposition approach in combination with initial bounds and different feasibility cuts for the Buffer Allocation Problem, which provides exact solutions while reducing the computation times substantially. Numerical experiments are carried out to demonstrate the performance and the flexibility of the proposed approaches. The numerical study reveals that the algorithm is capable to solve long lines with reliable and unreliable machines, including arbitrary distributions as well as correlations of processing times.

Time-dependent performance evaluation for loss-waiting queues with arbitrary distributions

This paper presents an analytical approach to evaluate queues with time-dependent, generally distributed inter-arrival times, generally distributed service times, and finite buffer capacities. A stationary backlog carryover (SBC) approach is developed to analyse the probability of blocking and other time-dependent performance measures. We further improve the general SBC approach by the analysis of load-dependent period lengths used in the approximation. The numerical study shows that this approach is very accurate for both transient and time-dependent loss-blocking systems.

Service-Level Variability of Inbound Call Centers

In practice, call center service levels are reported over periods of finite length that are usually no longer than 24 hours. In such small periods the service level has a large variability. It is therefore not sufficient to base staffing decisions only on the expected service level. In this paper we consider the classical M/M/s queueing model that is often used in call centers. We develop accurate approximations for the service-level distribution based on extensive simulations. This distribution is used for a service-level variability-controlled staffing approach to circumvent the shortcomings of the traditional staffing based on the expected service level.

Setting inventory levels of CONWIP flow lines via linear programming

This paper treats the problem of setting the inventory level and optimizing the buffer allocation of closed-loop flow lines operating under the constant-work-in-process (CONWIP) protocol. We solve a very large but simple linear program that models an entire simulation run of a closed-loop flow line in discrete time to determine a production rate estimate of the system. This approach introduced in Helber, Schimmelpfeng, Stolletz, and Lagershausen (2011) for open flow lines with limited buffer capacities is extended to closed-loop CONWIP flow lines. Via this method, both the CONWIP level and the buffer allocation can be optimized simultaneously. The first part of a numerical study deals with the accuracy of the method. In the second part, we focus on the relationship between the CONWIP inventory level and the short-term profit. The accuracy of the method turns out to be best for such configurations that maximize production rate and/or short-term profit.

Call Center Management in der Praxis

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