Service systems

Many customer service systems experience time-dependent arrival rates and numbers of servers (Thompson, 1993; Whitt, 2013). Surveys on time-depend-ent queueing models that are used for staffing decisions in service systems are provided by Green et al. (2007), Whitt (2007), Defraeye and Van Nieuwen-huyse (2011), and Defraeye and Van NieuwenNieuwen-huyse (2016). Hampshire and Massey (2010) integrate the performance analysis of time-dependent queue-ing systems in the optimization of multiple aspects of the communications industry. The applications can be categorized into the areas of telephone call centers, health care facilities, emergency services, service counters, and re-pair facilities.

Inbound telephone call centers are often characterized by a time-dependent arrival rate and a time-dependent number of agents (Gans et al., 2003). Where-as Sze (1984), Aguir et al. (2004), and Ridley et al. (2004) describe the perfor-mance evaluation of call centers, all other references cited in this paragraph concentrate on the development of staffing algorithms for call centers. Kole-sar and Green (1998) focus on the analysis of the peak hour in their staffing analysis. Most of the models for call centers apply queueing systems with Poisson arrivals and exponentially distributed service times (Andrews and Parsons, 1993; Kolesar and Green, 1998; Green et al., 2001, 2003; Koole and van der Sluis, 2003; Ridley et al., 2004; Dietz and Vaver, 2006; Atlason et al., 2008; Hampshire et al., 2009; Ingolfsson et al., 2010). Abandonments are considered by Feldman et al. (2008), Hampshire et al. (2009), Bertsimas and Doan (2010), Dietz (2011), and Kim and Ha (2012). Customers who reen-ter the system afreen-ter abandonment (retrials) are analyzed by Sze (1984) and Aguir et al. (2004). Sze (1984) considers abandonments as part of the ar-riving jobs that require a service time of zero. All of the models mentioned above consider systems with an infinite waiting room. In contrast, Mok and Shanthikumar (1987) consider a system with a limited waiting room. A call center that can be modeled as an M(t, n)/G/c queueing system with state-dependent balking is considered by Chassioti et al. (2014). Mok and Shan-thikumar (1987) consider a heterogeneous queueing system with two server classes, i.e., scheduled servers and standby servers that are used only if the queue exceeds a predetermined threshold. Different job classes and job class-dependent priorities are considered by Ridley et al. (2004) and Bertsimas and Doan (2010).

The request for medical services at health care facilities, such as emer-gency departments, can vary significantly over time (Bhattacharjee and Ray, 2014). Consequently, the number of medical personnel is often also time-dependent. Applications include the performance analysis of emergency fa-cilities (Collings and Stoneman, 1976; de Bruin et al., 2007), staffing in clin-ical wards (Agnihothri and Taylor, 1991; Gillard and Knight, 2014; Yom-Tov and Mandelbaum, 2014), and ambulance management (Singer and Donoso, 2008). Yom-Tov and Mandelbaum (2014) derive a model that includes re-entrant patients/repetitive service in clinical wards. Brahimi and Worthington (1991a) and Bennett and Worthington (1998) use the DTA to analyze out-patient appointment systems. The optimal out-patient mix with respect to out-patient service requirements is analyzed by Vanberkel et al. (2014).

Similarly, emergency services providers, such as police or fire fighters, face time-dependent service requests. Such systems are considered in the staffing and scheduling algorithm of Green et al. (2006). Green and Kolesar (1995) evaluate peak hour effects for emergency service systems. Bookbinder and Martell (1979) minimize the damage potential of forest fires by considering the allocation of available helicopters. Alfa and Margolius (2008) evaluate the queue of requests for police patrol cars, and Kolesar et al. (1975) and Ingolfsson et al. (2002) apply time-dependent queueing systems as part of scheduling algorithms for police patrol cars.

Service counters and facilities in airport terminals, such as check-in coun-ters, security checks, departure lounges, and baggage claim facilities, experi-ence time-dependent traveler arrivals. A detailed description of these applica-tions is provided in the survey by Toˇsi´c (1992). The approaches of Horonjeff (1969), Paullin and Horonjeff (1969), de Neufville and Grillot (1982), Wiras-inghe and Shehata (1988), WirasWiras-inghe and Bandara (1990), and de Barros and Tomber (2007) rely on the fluid approximation. Stolletz (2011) uses the SBC to analyze the performance of check-in counters. As another type of service counter with time-dependent arrivals, a fast food restaurant is studied by Kwan et al. (1988). Foote (1976) evaluates the performance of a drive-in banking facility with multiple lines involving jockeying. Kolesar (1984) ana-lyzes the expected number of waiting customers in front of automated teller machines to evaluate different layouts for a bank lobby. The staffing at border crossings in the form of a stationary congestion-based policy is considered by Zhang (2009). Liu and Wein (2008) derive a model to determine the number of necessary beds for the detention and removal of illegal aliens at border crossings.

The demand for repairs at repair facilities is also often time-dependent. The analysis of such systems provides insights into the required inventory level of spare parts over time. Jung (1993) analyzes a repair facility for expensive air-craft parts. Jung and Lee (1989a) and Lau and Song (2008) optimize stocking levels of spare parts in repair facilities in a military context. Buczkowski and Kulkarni (2006) use the explicit solution of an M(t)/G/∞ system to model the time-dependent number of items under warranty to determine the optimal funding of a warranty reserve.

All references reporting an application in the area of service systems are listed in Table 3.12. The second column (Emb.) shows that the performance eval-uation is often embedded within optimization algorithms, especially for call centers and emergency services. The third column indicates whether real-world data are used in the numerical study. The fluid approximation and methods based on stationary models are most frequently used.

Table 3.12: Applications in the area of service systems

Reference Emb. Real data Eval. method

Telephone call centers

Sze (1984) SIPP

Mok and Shanthikumar (1987) x x UR

Andrews and Parsons (1993) x x SIPP

Kolesar and Green (1998) x x SPHA

Green et al. (2001) x x SIPP

Green et al. (2003) x x SIPP

Koole and van der Sluis (2003) x x SIPP

Aguir et al. (2004) x FLUID

Ridley et al. (2004) x FLUID

Dietz and Vaver (2006) x SIPP

Atlason et al. (2008) x SIPP

Feldman et al. (2008) x INFSA, MOL

Hampshire et al. (2009) x FLUID

Bertsimas and Doan (2010) x x FLUID

Ingolfsson et al. (2010) x SIPP, UR

Dietz (2011) x x SIPP

Kim and Ha (2012) x x EXPL

Chassioti et al. (2014) x PSA

Health care

Collings and Stoneman (1976) EXPL

Agnihothri and Taylor (1991) x x SIPP

Brahimi and Worthington (1991a) x DTA

Bennett and Worthington (1998) x DTA

de Bruin et al. (2007) x SIPP

Singer and Donoso (2008) x SIPP

Gillard and Knight (2014) x CKE

Vanberkel et al. (2014) x x PSA

Yom-Tov and Mandelbaum (2014) x x MOL

Table 3.12: Applications in the area of service systems - continued

Reference Emb. Real data Eval. method

Emergency services

Kolesar et al. (1975) x x CKE, SIPP

Bookbinder and Martell (1979) x x CKE

Green and Kolesar (1995) SPHA

Ingolfsson et al. (2002) x x CKE, SIPP

Green et al. (2006) x x SIPP

Alfa and Margolius (2008) x DTA

Service counters

Horonjeff (1969) x FLUID

Paullin and Horonjeff (1969) x FLUID

Foote (1976) x SIPP

de Neufville and Grillot (1982) x FLUID

Kolesar (1984) x SIPP

Kwan et al. (1988) x x SIPP

Wirasinghe and Shehata (1988) x FLUID

Wirasinghe and Bandara (1990) x x FLUID

Thompson (1993) x SIPP

de Barros and Tomber (2007) x FLUID

Liu and Wein (2008) x MOL, PSA

Zhang (2009) SIPP

Stolletz (2011) SBC

Repair facilities

Jung and Lee (1989a) x DIFF, FLUID

Jung (1993) x CKE

Buczkowski and Kulkarni (2006) x EXPL

Lau and Song (2008) x SDA