1. Introduction
Many hospital systems (e.g., Lutheran Medical Center and UW Health) have implemented interventions known as “split flow” models to improve patient flow in the Emergency Departments (EDs) (cf., [Reference Kamali, Tezcan and Yildiz21,Reference Konrad, DeSotto, Grocela, McAuley, Wang, Lyons and Bruin29,Reference Zayas-Cabán, Xie, Green and Lewis49,Reference Zayas-Caban, Xie, Green and Lewis50]). Unlike a typical ED, a split flow model stations an advanced practice provider (APP) rather than a nurse at patient intake (i.e., before or during triage). This provider briefly sees all walk-in patients and may initiate the care of all patients by placing lab, imaging, and medication orders (i.e., phase one service). The provider then stratifies patients, keeping those who do not require a traditional ED bed in a fast track or similar area (i.e., phase two service) and moving the rest to a queue for a traditional bed. Importantly, the same care provider is continuously switching between both phases of service, and while patients rarely leave before being seen while awaiting phase one service (e.g., triage), some may abandon the system before receiving final treatment.
Motivated by the ED split flow model, we consider the assignment of servers between two phases of service in a two-stage tandem queue. This and related models have been studied in the case when one or more job classes require service at multiple service stations in tandem by one or more servers (cf., [Reference Ahn and Righter1–Reference Argon and Tsai8,Reference Duenyas, Gupta and Olsen15,Reference Iravani, Posner and Buzacott17,Reference Işık, Andradóttir and Ayhan18,Reference Johri and Katehakis20,Reference Katayama25,Reference Kaufman, Ahn and Lewis27,Reference Nelson35,Reference Pandelis37,Reference Schiefermayr and Weichbold41,Reference Sidi, Levy and Fuhrmann42,Reference Van Oyen, Gel and Hopp46,Reference Zayas-Cabán, Xie, Green and Lewis49]). Servers decide where to allocate efforts to optimize performance criteria on quality of service and/or congestion. Although widely studied, many of these models assume that jobs have unlimited patience and thus willing to wait indefinitely. However, there are practical and important situations, such as the ED, where jobs abandon the system before service. Allowing for abandonments, we analyze this scheduling problem to find new control strategies that generalize to broader situations.
In the present study, we assume Poisson arrival streams to each phase of service with infinite waiting capacity, and one or more flexible servers that can serve arriving jobs having service times that are identically distributed and are independent, and are independent of inter-arrival times. We assume that no setup is required for the server to switch from processing jobs at one station to another. In addition, we assume that jobs can abandon from each phase of service. The goal is to provide server assignment policies to minimized expected discounted or average costs over an infinite horizon where each job class has linear holding costs and lump-sum abandonment costs.
To analyze this decision-making scenario, we first assume exponentially distributed service and abandonment times and use Markov decision processes (MDPs). For the multi-server model, we show that costs are nondecreasing in the number of customers in each phase of service, and that there exists a non-idling policy that is optimal so long as service preemption is allowed. We then show that if the number of customers at each queue exceeds the number of servers, there exists an optimal control policy that does not split the servers so long as service preemption is allowed. We then consider the single-server model and identify a set of conditions under which static priority rules are optimal. Outside these conditions, the optimal policy may be a complicated state-dependent policy. We thus consider a class of threshold policies which we evaluate in a two-part simulation study.
In the first part, we consider the single-server Markovian model and compare the performance of these threshold policies to the priority rules in the cases where we have proved that the priority rules are optimal. This is done to benchmark the proposed threshold policies. We then compare their relative performance in a discrete-event simulation for a wider range of parameter values. In the second part, we evaluate these threshold policies when abandonments during service and preemption are not allowed and service and abandonment times are no longer exponential.
The rest of the paper is organized as follows: Section 2 contains a summary of the literature related to our work. Section 3 describes in detail the model we consider, the MDP formulation of the server allocation problem, and some preliminary results that will be used throughout the paper. Section 4 contains our main analysis of static priority rules. In Section 5, we propose several allocation policies based on our analytical results and compare them numerically. We conclude with a discussion in Section 6.
2. Brief summary of the related literature
This study lies at the intersection of three areas: scheduling in tandem queueing systems, performance analysis of service policies in tandem queues, and MDPs with unbounded transition rates.
To our knowledge, the only other studies combining these three areas are [Reference Zayas-Cabán, Xie, Green and Lewis49,Reference Zayas-Caban, Xie, Green and Lewis50]. In Zayas-Cabán et al. [Reference Zayas-Cabán, Xie, Green and Lewis49], policies for a two-stage tandem queueing model with abandonments and rewards accrued after service completion were analyzed using a continuous-time Markov decision process (CTMDP), motivated by the Triage and Treat Release Program at the Lutheran Medical Center in New York. In a follow-up study, Zayas-Cabán et al. [Reference Zayas-Caban, Xie, Green and Lewis50] consider the same two-phase stochastic service system but where customers may only abandon the second phase of service. They introduce a class of policies they term 
$K$-level threshold policies, which prioritize phase 2 service unless there are 
$K$ or more jobs in phase one service. Sufficient conditions are provided to ensure these policies yield a stable system. A heuristic is presented for choosing 
$K$ in systems with abandonments. They analyze the performance of these heuristics in a simulation study. The present study differs from [Reference Zayas-Cabán, Xie, Green and Lewis49] and [Reference Zayas-Caban, Xie, Green and Lewis50] by allowing arrivals to the second phase of service and by considering holding costs and abandonment costs (and not rewards). Holding and abandonment costs imply that cost rates are unbounded, which require stronger conditions to hold in order for the optimality equations to have a solution (cf., [Reference Blok and Spieksma13]). Furthermore, holding cost rates imply that stronger conditions are needed for the optimality of prioritizing station 2 to hold. The bounds for the value function that were obtained in [Reference Zayas-Cabán, Xie, Green and Lewis49] to imply that the optimality of static priority rules cannot be similarly obtained in the present study.
There is extensive literature on dynamic assignment of servers to different phases of service without abandonments (cf., [Reference Ahn and Righter1–Reference Argon and Tsai8,Reference Duenyas, Gupta and Olsen15,Reference Iravani, Posner and Buzacott17,Reference Işık, Andradóttir and Ayhan18,Reference Johri and Katehakis20,Reference Katayama25,Reference Kaufman, Ahn and Lewis27,Reference Nelson35–Reference Pandelis37,Reference Schiefermayr and Weichbold41,Reference Sidi, Levy and Fuhrmann42,Reference Van Oyen, Gel and Hopp46,Reference Zayas-Cabán, Xie, Green and Lewis49]). Nelson [Reference Nelson35], for instance, presents the optimality of priority rules akin to the classic c-
$\mu$ rule but for a tandem system without abandonments in the context of a labor assignment problem over a finite horizon using the Pontryagin maximum principle. There is also extensive literature on performance analysis of single-server tandem queues without abandonments, which provide analysis techniques for determining stability conditions of different allocation policies such as priority rules (cf., [Reference Katayama22–Reference Katayama24,Reference Katayama and Kobayashi26,Reference Nair34,Reference Taube-Netto43]). We refer the reader to [Reference Zayas-Cabán, Xie, Green and Lewis49,Reference Zayas-Caban, Xie, Green and Lewis50] for a recent review of this literature. More recently, Wang et al. [Reference Wang, Abouee-Mehrizi, Baron and Berman47] and Rastpour et al. [Reference Rastpour, Ingolfsson and Sandıkçı39] analyze multi-server Markovian queues with abandonments. Viewed as a level-dependent quasi-birth and death process, Wang et al. [Reference Wang, Abouee-Mehrizi, Baron and Berman47] analyze a multi-server tandem queue with abandonments where the second phase of service has a finite number of servers.
Lastly, we contend with a CTMDP with unbounded rates as a consequence of abandonments (cf., [Reference Bhulai, Brooms and Spieksma10–Reference Down, Koole and Lewis14,Reference Guo and Hernández-Lerma16,Reference Koçağa and Ward28,Reference Legros, Jouini and Koole32,Reference Liang and Kulkarni33,Reference Prieto-Rumeau and Hernández-Lerma38,Reference Saghafian, Hopp, Van Oyen, Desmond and Kronick40,Reference van Dijk44,Reference Van dijk45,Reference Zayas-Cabán and Ahn48,Reference Zayas-Cabán, Xie, Green and Lewis49]). In particular, it extends the cost model considered by Down et al. [Reference Down, Koole and Lewis14] by allowing phase one service completions to join phase two service. Throughout the paper, and whenever possible, we compare our results to those obtained in Down et al. [Reference Down, Koole and Lewis14]. Although the remaining above-cited papers consider different models than the one considered here, they also provide approaches for how to analyze optimal controls for problems with unbounded rates. We remark that alternative approaches to CTMDPs have been used to contend with scheduling with abandonments (cf., [Reference Atar, Giat and Shimkin9,Reference Larranaga, Ayesta and Verloop30,Reference Larrañaga, Ayesta and Verloop31]). For example, Atar et al. [Reference Atar, Giat and Shimkin9] consider the 
$K$-competing queues problem with many servers and introduce the 
$h \mu / \beta$ rule, which prioritizes the queue with the highest index 
$h_c \mu _c / \beta _c$, showing that it is asymptotically optimal in the so-called overloaded regime, as the number of servers tends to infinity.
3. Dynamics and control formulation
Suppose customers, or jobs, arrive to station 1 (2), or phase 1 (2) service, or queue 1 (2), of a tandem service system according to a Poisson process of rate 
$\lambda _1$ (
$\lambda _2$) and immediately join the first (second) queue. Once the customer joins station 1 (2), their station 1 (2) patience time and service requirements are generated; the distribution of the former and latter will be first assumed to be exponential with rate 
$\beta _1$ (
$\beta _2$) and 
$\mu _1$ (
$\mu _2$), respectively. If the customer does not complete station 1 (2) service before the abandonment time ends, a lump-sum cost 
$K_1$ (
$K_2$) is charged and the customer leaves the system without receiving service at station 1 or 2 (2). If the customer receives service at station 1 then, independently of the service time and arrival process, with probability 
$p \in [0,1]$, the customer joins the queue at station 2. With probability 
$q:=1-p$, the customer leaves the system forever.
There is a nonnegative holding cost 
$h_{c}$ per job per unit time incurred for holding a customer in station 
$c \in \{1,2\}$. There are 
$N \geq 1$ servers, each of which can be assigned to either station. We seek a non-anticipating policy that describes where to place the server based on the current state and potentially the history of states and actions taken. Within each station, the service discipline is first come first served (FCFS), and we consider both when service can and cannot be preempted and when abandonments of jobs in process are and are not allowed. Once a customer completes service at station 2, they leaves the system forever.
Fix a non-anticipating policy 
$\pi$, let 
$\{\sigma _n^{\pi }, n \geq 1\}$ denote the sequence of event times that includes arrivals, abandonments, and potential service completions, and let 
$\mathbb {Z}^{+}$ denote the set of nonnegative integers. For this analysis, we first assume that service can be preempted and abandonments during service are allowed. Throughout the paper, and whenever possible, we highlight when these restrictions can be relaxed. When service can be preempted and abandonments during service are allowed, the state space is 
$\mathbb {X} := \{(x_1,x_2) \,| \,x_2, x_2 \in \mathbb {Z}^{+}\}$, where 
$x_1$ (resp. 
$x_2$) denotes the number of customers at station 1 (resp. 2) and the set of available actions at state 
$x \in \mathbb {X}$ is 
$A(x) = \mathbb {A} := \{(y_1, y_2)\,|\, y_1, y_2 \in \mathbb {Z}^{+}, y_1 + y_2 \leq N \}$, where 
$y_1$ (
$y_2$) represents the number of servers assigned to station 1 (2).Footnote 1 For 
$\alpha \gt 0$, the finite horizon (of length 
$t$), 
$\alpha$-discounted expected cost for a non-anticipating policy 
$\pi$ is given by 
$v_{t,\alpha }^{\pi }(x) \equiv \mathbb {E}_{x}^{\pi } [\sum _{n=0}^{N(t,\pi )} e^{-\alpha \sigma _n} k(X_n,a_n)] + \int _0^{t} e^{-\alpha s} \mathbb {E}_{x}^{\pi } [ \sum _{c=1}^{2} h_{c} Q^{\pi }_{c}(s)) ]\,ds$, where 
$Q^{\pi }_{c}(s)$ denotes the customer class 
$c \in \{1,2\}$ queue length process at time 
$s \geq 0$, and 
$X_n$ and 
$a_n$ represent, respectively, the state of the system and the type of event seen at the time of the 
$n$th decision. The function 
$k(\cdot, \cdot )$ denotes the fixed cost; that is, if 
$\sigma _n$ denotes a station 
$c \in \{1,2\}$ customer abandonment, then 
$k(X_n,a_n) = K_{c}$ and it is zero otherwise. For fixed 
$x \in \mathbb {X}$, the infinite horizon discounted expected cost under policy 
$\pi$ is 
$v_{\alpha }^{\pi }(x) \equiv \lim _{t \rightarrow \infty } v_{t,\alpha }^{\pi }(x)$. The long-run average cost rate is 
$\rho ^{\pi }(x) \equiv \limsup _{t \rightarrow \infty } v_{t,0}^{\pi }(x)/t$.
4. Dynamic control
We present two results that are used throughout when abandonments during service and preemption are allowed. The first is the monotonicity of the value functions. The second says that there is an optimal policy that does not idle the servers whenever there are customers waiting so long as service can be preempted. The latter is used to simplify the optimality equations. In the interest of brevity, we omit the proofs.
Proposition 4.1 The following hold:
1. For all
$x=(x_1,x_2) \in \mathbb {X}$
where$$v_{\alpha}(x+e_{c}) \geq v_{\alpha}(x), \quad c = 1, 2$$
$e_{c}$ denotes the $c$th standard basis vector in $\mathbb {R}^{2}$ ($c=1,2$). Similarly, if $(g,w)$ is a solution to the average cost optimality equations (defined below), the above statements hold with $v_{\alpha }$ replaced with $w$.2. Under both the
$\alpha$-discounted cost and the average cost criterion, there exists a (Markovian) non-idling policy that is optimal.
We remark that a similar result to the first part of Proposition 4.1 holds when abandonments during service and service preemption are not allowed, but a similar result to the second part of Proposition 4.1 only holds when abandonments during service are not allowed. When service preemption is not allowed, it may be optimal for the server to stay idle.
4.1. Optimality equations
For functions 
$f:\mathbb {X} \rightarrow \mathbb {R}$, let 
$f(x_1,x_2):= f((x_1, x_2))$ for 
$(x_1,x_2) \in \mathbb {X}$. The operator 
$T$ acting on functions 
$f:\mathbb {X} \rightarrow \mathbb {R}$ is defined as follows.
\begin{align*} Tf(x_1,x_2)& := x_1 (h_1 + \beta_1 K_1) + x_2 (h_2 + \beta_2 K_2) + \lambda_1 f(x_1+1,x_2) + \lambda_2 f(x_1,x_2+1) \\ & \quad + x_1\beta_1f(x_1-1,x_2) + x_2\beta_2 f(x_1, x_2-1) \\ & \quad - (\lambda_1 + \lambda_2 + x_1\beta_1 + x_2\beta_2)f(x_1,x_2) \\ & \quad + \min_{a_1 \in \{0,1,\ldots,N\}}\{\min\{x_1,a_1\}\mu_1 [ pf(x_1-1, x_2+1) + qf(x_1-1,x_2) - f(x_1,x_2)] \\ & \quad + \min\{x_2,N-a_1\}\mu_2[f(x_1, x_2-1) - f(x_1,x_2)]\}. \end{align*}
Theorem 4.2 For any 
$\alpha \gt 0$, the following statements hold.
1. The value function
$v_\alpha$ satisfies the discounted cost optimality equations (DCOE), that is,
$$\alpha v_\alpha(x_1,x_2) = Tv_\alpha(x_1,x_2), \quad (x_1,x_2) \in \mathbb{X}.$$
2. There exists a deterministic stationary
$\alpha$-optimal policy.3. Any policy satisfying the maximum in the DCOE is
$\alpha$-optimal.
Proof. See Online Appendix in Supplementary material.
Theorem 4.3 If any of the following mutually exclusive conditions holds
•
$\min \{\beta _1,\beta _2\}\gt0$; or•
$\beta _1\gt0$, $\beta _2=0$, and if ${\lambda _2}/{\mu _2}\lt 1$ for the multi-server model, or, if $\lambda _1 \cdot ({1}/{\pi _0(\mu _1+\beta _1)} + {p(1-P(Ab)}/{\mu _2}) + {\lambda _2}/{\mu _2} \lt 1$ for the single-server model, where $\pi _0$ is long-run fraction of time that station 2 is empty under the non-idling policy that prioritizes station 2 when $\beta _2=0$ and $P(Ab)=\beta _1/(\mu _1+\beta _1)$ ; or•
$\beta _1=0$, $\beta _2\gt0$ and ${\lambda _1}/{\mu _1} \lt 1$,
then the following hold:
1. There is a constant
$g \in \mathbb {R}$ and a function $w:\mathbb {X} \rightarrow \mathbb {R}$ that satisfy the average cost optimality equations (ACOE), that is,
$$g = Tw(x_1,x_2), \quad (x_1,x_2) \in \mathbb{X}.$$
2. There exists a deterministic stationary average-optimal policy.
3. Any policy satisfying the maximum in the AROE is average-optimal, and
$g = \rho (x_1,x_2)$ for all $(x_1,x_2) \in \mathbb {X}$.
Proof. See Online Appendix in Supplementary material.
We remark that similar results to Theorems 4.2 and 4.3 hold when service preemption and abandonments during service are not allowed.
4.2. Dynamic control
For the multi-server model, we have the following result, which is akin to Proposition 3.1 in [Reference Zayas-Cabán, Xie, Green and Lewis49] on the allocation of servers when the number of customers is sufficiently high. The proof is similar to Proposition 3.1 in [Reference Zayas-Cabán, Xie, Green and Lewis49] and is therefore omitted.
Proposition 4.4 For the non-collaborative model we are considering, if the number of customers at each queue exceeds the number of servers, there exists a discounted cost optimal control policy that allocates all servers to one phase or the other (i.e., servers are not split between the two queues). Similarly, the result holds in the average cost case when the average cost optimality equations have a solution.
We remark that if we assume that when more than one server is working at a station, their rates are additive, then, for the multi-server model with abandonments during service and service preemption, there is a discounted cost and average cost optimal policy that does not split the servers between the two phases of service. We also note that Proposition 4.4 does not hold when service preemption is not allowed. Proposition 4.4 implies that when abandonments during service and preemption are allowed, we can restrict attention to policies that always allocate all servers to one station or the other when both 
$x_1,x_2 \geq N$ (making the service rate 
$N \mu _1$ or 
$N \mu _2$). Furthermore, the second part of Proposition 4.1 implies that we should keep as many servers busy as possible in states such that 
$x_1+x_2 \leq N$ when service preemption is allowed. However, we were unable to further characterize the optimal policy in states with 
$x_1$ or 
$x_2$ (but not both) greater than 
$N$ nor how to choose priorities when 
$x_1,x_2 \geq N$. In the following, we discuss under what conditions static priority policies are optimal in the single-server model (see Section 4.3) and revisit the multi-server model in our numerical study (Section 5).
4.3. The single-server proxy
In this section, we restrict attention to the single-server model (
$N=1$) and provide conditions under which one particular phase of service should be prioritized.
Theorem 4.5 Under the 
$\alpha$-discounted cost criterion, it is optimal to serve at station 2 whenever station 2 is not empty if either one of the following two sets of conditions hold:
1.
$\beta _2=0$ and $\mu _1[h_1+\beta _1K_1-p h_2] \leq \mu _2h_2$; or2.
$\mu _1=\mu _2$, $\beta _1-\beta _2-\mu _2 \geq 0$, and $h_1 + \beta _1K_1-p(h_2 + \beta _2K_2) \leq h_2 + \beta _2K_2$.
If, in addition, 
$\lambda _1 \cdot ({1}/{\pi _0(\mu _1+\beta _1)} + {p (1-P(Ab)}/{\mu _2}) + {\lambda _2}/{\mu _2}\lt 1$, conditions 1 or conditions 2 imply that serving station 2 when station 2 is not empty is also average cost optimal.
Proof of 4.5 when conditions 1 hold. Proof of 4.5 when conditions 1 hold
The proof is given for the discounted expected cost model. The proof of the long-run average cost case is similar. Note that the optimality equations imply that it is optimal to prioritize a class 
$2$ customer in state 
$(x_1,x_2)$ with 
$x_1,x_2 \geq 1$ when
\begin{equation} \mu_1[p v_{\alpha}(x_1-1,x_2+1) + q v_{\alpha}(x_1-1,x_2) - v_{\alpha}(x_1,x_2)] + \mu_2[v_{\alpha}(x_1,x_2)-v_{\alpha}(x_1,x_2-1)] \geq 0. \end{equation}
We show (4.1) via a sample path argument. Fix 
$x_1,x_2 \geq 1$ and start five processes on the same probability space. Processes 1–5 begin in states 
$(x_1-1,x_2+1), (x_1-1,x_2), (x_1,x_2)$, 
$(x_1,x_2)$, and 
$(x_1,x_2-1)$, respectively. Processes 1, 2, and 4 use stationary optimal policies, which we denote by 
$\pi _1$, 
$\pi _2$, and 
$\pi _4$, respectively. In what follows, we show how to construct (potentially sub-optimal) policies for Processes 3 and 5 which we denote by 
$\pi _3$ and 
$\pi _5$, so that
\begin{equation} \mu_1[p v_{\alpha}^{\pi_1}(x_1-1,x_2+1) + q v_{\alpha}^{\pi_2}(x_1-1,x_2) - v_{\alpha}^{\pi_3}(x_1,x_2)] + \mu_2[v_{\alpha}^{\pi_4}(x_1,x_2)-v_{\alpha}^{\pi_5}(x_1,x_2-1)] \geq 0. \end{equation}
Since 
$\pi _3$ and 
$\pi _5$ are potentially suboptimal, (4.1) follows from (4.2). In what follows, discounting is suppressed without any loss of generality.
Observe that starting from (4.2), the costs incurred until the next event are 
$(\mu _2 h_2- \mu _1[h_1+\beta _1K_1-p h_2])t_1 \geq 0$, where 
$t_1$ is the time of the next event and the inequality is due to the assumption that 
$\mu _2h_2 \geq \mu _1[h_1+\beta _1K_1-ph_2]$. Moreover, if the relative position (as measured by the current states) of the five processes at the next event remains the same, then we may relabel the initial states and continue from the beginning of the argument. This occurs when any of the uncontrolled events occur that are seen by all five processes (i.e., an arrival or an abandonment at station 
$i$ and 
$x_{i}\gt1$) . It also occurs when the next event is a service completion and when 
$\pi _1$, 
$\pi _2$, and 
$\pi _4$ serve the same customer class 
$k \in \{1,2\}$ by letting 
$\pi _3$ and 
$\pi _5$ also serve the same customer class 
$k$ customer provided there is one or more customer class 
$k$ customer in all five processes. Consider now the other cases.
Case 1 Customer abandonments
If 
$x_1=1$ and the first event is a class 1 abandonment in Processes 3–5 only (with probability 
${\beta _1}/{(\lambda _1 + \lambda _2 + \mu _1 + \mu _2 + x_1 \beta _1)}$), after which all processes follow an optimal control, it follows that the remaining costs on the left side of (4.2) (with the probability of this event in the expression suppressed) are
\begin{align} & (\mu_2h_2-\mu_1[h_1+\beta_1K_1-ph_2])t_1 + p \mu_1[v_{\alpha}(x_1-1,x_2+1)-v_{\alpha}(x_1-1,x_2)] \nonumber\\ & \quad+ \mu_2[v_{\alpha}(x_1-1,x_2)-v_{\alpha}(x_1-1,x_2-1)]. \end{align}The terms in this last expression above are nonnegative as a consequence of Proposition 4.1.
There are seven cases left to consider corresponding to service completions all with algebra that is directly analogous. Complete details are available in the Online Appendix in Supplementary material. It follows that, in every case save one (Case 1), we may relabel the states and continue. Thus, to obtain the result, we wait until Case 1 occurs.
Proof of 4.5 when conditions 2 hold. Proof of 4.5 when conditions 2 hold
See Online Appendix in Supplementary material.
We make several observations about Theorem 4.5 and its associated conditions. First, the policy that prioritizes station 2 is nonpreemptive. Second, if 
$\min \{\beta _1,\beta _2\} \gt0$, then 
$\mu _1 [h_1+\beta _1 K_1-p(h_2+\beta _2K_2)] \leq \mu _2 (h_2+\beta _2 K_2)$ and 
${\mu _1 [h_1+\beta _1 K_1-p(h_2+\beta _2K_2)]}/{\beta _1} \leq {\mu _2 (h_2+\beta _2 K_2)}/{\beta _2}$ do not guarantee the optimality of prioritizing station 2, as examples in Section 5 illustrate. Third, the stability condition 
$\lambda _1 \cdot ({p(1 - \Pr (\text {Ab}))}/{\pi _0 \mu _1} + {1}/{\mu _2}) + {\lambda _2}/{\mu _2} \lt 1$ is derived by computing the fraction of customers arriving to station 1 that enter station 2 and then computing the fraction of time the server works at station 1 (under the prioritize station 2 policy) (see the Online Appendix in Supplementary material for complete details). Fourth, the second condition in 2 says that the station 1 abandonment rate 
$\beta _1$ is higher than the total departure rate from station 2, 
$\beta _2+\mu$. If 
$p=0$, then the system is the same as the one considered by Down et al. [Reference Down, Koole and Lewis14]. In this case, the term corresponding to 
$(\beta _2-\beta _1-\mu )p$ in subcase 1.2 in the Online Appendix in Supplementary material is 
$0$ so that the same proof of 4.5 when conditions 1 holds, yields that 
$h_2 + \beta _2K_2 \geq h_1+ \beta _1K_1$ and 
$\beta _1 \geq \beta _2$ imply that it is optimal to prioritize station 2, in agreement with Theorem 3.5 of Down et al. [Reference Down, Koole and Lewis14]. Fifth, one may conjecture that Theorem 4.5 extends to the case when abandonments during service are not allowed, but this turns out not to be so simple since the allocation decision impacts the abandonment rate and the current per unit holding cost, suggesting that additional conditions may be required.
Theorem 4.6 Under the 
$\alpha$-discounted cost criterion, it is optimal to serve at station 1 whenever station 1 is not empty if one of the following two sets of conditions hold:
1.
$\beta _1=0$ and $\mu _2[h_2 + \beta _2K_2] \leq \mu _1[h_1-p(h_2 + \beta _2K_2)]$; or2.
$\mu _1=\mu _2$, $\beta _2\geq \beta _1$, and $h_2 + \beta _2K_2 \leq h_1 + \beta _1K_1-p(h_2 + \beta _2K_2)$.
If, in addition, 
${\lambda }/{\mu _1} \lt 1$, conditions 1 or 2 imply that serving station 1 when station 1 is not empty is also average cost optimal.
Proof of 4.6. Proof of 4.6.
See Online Appendix in Supplementary material.
We note that when 
$\min \{\beta _1,\beta _2\} \gt0$, 
$\mu _1 [h_1+\beta _1 K_1-p(h_2+\beta _2K_2)] \geq \mu _2 (h_2+\beta _2 K_2)$ or 
${\mu _1 [h_1+\beta _1 K_1-p(h_2+\beta _2K_2)]}/{\beta _1} \geq {\mu _2 (h_2+\beta _2 K_2)}/{\beta _2}$ do not guarantee that it is always optimal to prioritize station 1 except to avoid unforced idling, as examples in Section 5 illustrate. Furthermore, 
${\lambda _1}/{\mu _1} \lt 1$ is the stability condition for an M/M/1 queue and is sufficient for positive recurrence of all states under the prioritize station 1 policy (see the Online Appendix in Supplementary material for complete details). Lastly, the conditions 2 are the same conditions that guarantee that prioritize station 1 is optimal in Down et al. [Reference Down, Koole and Lewis14], which is a special case of our model when 
$p=0$.
5. Numerical study
5.1. Rationale
This section affords us the opportunity to address shortcomings from our analytical analysis. One shortcoming is that our model relies on unrealistic assumptions for systems of interest (i.e., one server, abandonments during service or preemption, and exponential service and abandonment times). Furthermore, the properties of optimal controls remain elusive in many scenarios. In the single-server Markovian model with preemption and abandonments during service, one might conjecture that if it is optimal to serve in station 
$1$ (
$2$) in state 
$(x_1,x_2)$, then it is also optimal to serve at station 
$1$ (
$2$) when there are more customers in station 
$1$ (
$2$) in state 
$(x_1+1,x_2)$ (
$(x_1,x_2+1$)). To show such a result, standard approaches include sample path arguments or a smoothed rate truncation approach. For either approach, abandonments lead to having to show several inequalities, such as submodularity, of the value function for the discounted cost problem and the relative value function for the long-run average cost problem (cf., [Reference Bhulai, Brooms and Spieksma10]). Unfortunately, submodularity remains elusive when the number of jobs at each station is zero; that is, at the boundary of the state space. Despite this, the optimal server allocation policy might still have a threshold or switching curve structure, but proving such a result without submodularity or convexity is difficult. Because the optimal server allocation policy is difficult to discern, we propose and evaluate a class of threshold heuristics that initially assign priority to one phase of service and then switch between priorities based on a state-dependent rule.
Our numerical study has four main goals. The first is to quantify the relative performance of heuristic allocation policies with respect to average costs. The second is to quantify the degree to which (non-priority) threshold policies outperform strict priority policies. The third is to determine how relative performance is affected by changes in the coefficient of variation for the service and abandonment times and changes in
$$\lambda_1\cdot\left(\frac{1}{\mu_1+\beta_1} + \frac{p}{\mu_2+\beta_2}\right) + \frac{\lambda_2}{\mu_2+\beta_2},$$
which we use as a proxy for the system load. This proxy load corresponds to the average service time for a job that is processed in both phase 1 and then phase 2 service under the prioritize station 2 policy when preemption and abandonments during service are allowed. The fourth is to determine the degree to which guidelines, like the classic c-
$\mu$ rule, correctly identify the best policy.
5.2. Overview
Our numerical study presents a discrete-event simulation. To simplify exposition, the main text focuses on a multi-server model without preemption and abandonments during service and without exponential service and abandonment times. We vary model parameters to recover guidelines for when one policy might be preferred over another. Simulation results are presented in the Online Appendix in Supplementary material for a single-server Markovian model that allows preemption and abandonments during service and that assumes service and abandonment times are exponentially distributed. The Online Appendix in Supplementary material also benchmarks (non-priority) threshold policies, with respect to long-run average costs, in situations when we know prioritizing a particular phase of service is optimal.
5.3. Heuristic allocation policies
Four types of policies are considered. Each policy is specified to prevent idling, that is, when there are not enough jobs to serve at one phase, providers work at the other phase. When there is enough work to do, we do not assume that providers are split between phases. The first type of policy, denoted by P1(
$n$), prioritizes phase 
$1$ service until there are 
$n$ total customers in the system after which phase 
$2$ service is prioritized until phase 
$2$ is emptied. When phase 
$2$ is empty, the policy is reset and phase 
$1$ is prioritized again. When 
$n=\infty$, policy P1(
$\infty$) is the priority rule (denoted simply by P1) that always prioritizes phase 
$1$ service over phase 
$2$.
In a reciprocal manner, we define a second type of policy P2(
$n$) to prioritize phase 
$2$ service until there are 
$n$ total customers in the system after which phase 
$1$ service is prioritized until phase 
$1$ is emptied. When phase 
$1$ is empty, the policy is reset and phase 
$2$ is prioritized again. When 
$n=\infty$, policy P2(
$n$) is simply the priority rule for phase 
$2$ (denoted by P2). Another policy considered, denoted by Exh, is an exhaustive policy. It prioritizes one phase of service until that phase is empty after which it switches to the other phase until that phase is empty. It continues to switch between phases, emptying each phase in turn before switching to the other. The last policy considered, denoted by Inc, prioritizes whichever phase of service has more customers. It is considered an increasing policy, because the threshold of phase 
$2$ customers for switching as a function of phase 
$1$ customers is an increasing function.
Since P1(
$n$) and P2(
$n$) prioritize different phases of service, we explore the degree to which one policy outperforms another when the inequality
\begin{equation} \mu_1 h_1 \leq \mu_2 h_2 \end{equation}
holds as well as when
\begin{equation} \mu_1(h_1+\beta_1 K_1-p(h_2 + \beta_2 K_2)) \leq \mu_2(h_2 + K_2 \beta_2) \end{equation}
holds. The former inequality reflects the classic c-
$\mu$ rule for deciding when to prioritize one phase of service over another when there are no abandonments and the latter inequality reflects the analog of the c-
$\mu$ rule when you account for abandonments and are part of the sufficient conditions we provided to guarantee the optimality of a policy that prioritizes phase 1 (policy P1) or phase 2 (policy P2).
5.4. Parameters
Parameters are summarized in Table 1 for the simulation. Parameters were chosen to capture situations when the optimal policy remains elusive. Given the importance of the classic c-
$\mu$ inequality (Eq. (5.1)) and its extended version (Eq. (5.2)), parameters were selected and varied to both satisfy and violate these situations. It is without any loss of generality that we can fix one cost and one rate. So, the abandonment cost 
$K_2$ at phase 
$2$ is fixed at 1 and the arrival rate 
$\lambda _1$ at phase 1 is fixed at 9. Assuming a time unit of hours, we simulated the system for each parameter set over a simulated time horizon of 5 years after a 5 year warm-up period and then performed 50 replications of this simulation. Average costs were averaged over the time horizon and then over the replications.
Table 1. Parameters used for the simulation.
Parameters have a similar interpretation as the single-server Markovian model, with the following exceptions. First, we fixed the number of workers to be 3. Second, abandonment and service times were modeled as Gamma random variables as opposed to exponential random variables. Gamma shape parameters ranged from 
$1/2$ or 
$3$, yielding random times that have standard deviations larger than their mean and smaller than their mean, complementing exponential random times, which have standard deviations equal to their mean. Coefficient of variations (cv) were respectively 1.4 and 0.6 for the two shape values. Last, parameters 
$\mu _1$, 
$\mu _2$, 
$\beta _1$, and 
$\beta _2$ refer to average rates, which meant that the rate parameters for the gamma distributions needed to be 
$\mu _1$, 
$\mu _2$, 
$\beta _1$, 
$\beta _2$ scaled by the corresponding shape parameter.
We first explored parameter space using a full factorial design of six parameters (
$\lambda _2$, 
$\mu _1$, 
$\mu _2$, 
$\beta _1$, 
$\beta _2$, 
$p$, and cv); each parameter had two levels corresponding to the lowest and highest value in the parameter range listed in Table 1. For each of these 128 sets of parameters, we then sampled 10,000 sets of costs (
$h_1$, 
$h_2$, 
$K_1$) uniformly from the parameter range listed in Table 1. We then systematically varied parameters while keeping fixed (unless otherwise specified) 
$\lambda _1=9$, 
$\lambda _2=0$, 
$\mu _1=\mu _2=8$, 
$p=\beta _1=\beta _2=h_1=h_2=K_2=1$, and 
$K_1=2$. Service rates 
$\mu _1$ and 
$\mu _2$ were systematically varied, followed by abandonment rates 
$\beta _1$ and 
$\beta _2$, holding cost rates 
$h_1$ and 
$h_2$, and arrival rate 
$\lambda _2$ and joining probability 
$p$.
5.5. Results
We first report relative performance of the policies over 1,280,000 samples of parameter space for the multi-server model (Table 2). Policy P2 had lowest average costs for 64.1% of these samples compared to 19.2% of samples for P1, 5.7% for P1(5), 5.2% for P2(5), 5.0% for Inc, and less than 1% for Exh. Following P2 is even better than using the classic c-
$\mu$ inequality (Eq. (5.1)) to guide when to use P2 over P1, the latter being best for only 59.1% of samples compared to the aforementioned 64.1% for P2. Following P2, however, is slightly worse than using the extended version of the c-
$\mu$ inequality (Eq. (5.2)) to guide when to use P2 over P2, which is best for 65.2% of samples.
Table 2. Percent samples for which given policy yields lowest average costs as a function of service rates (
$\mu _1$ and 
$\mu _2$) and approximate load under P2.
Samples were stratified by service rates 
$\mu _2$ and 
$\mu _1$ (Table 2). When 
$\mu _2$ is high and 
$\mu _1$ is low, policy P2 is, as expected, best for 86.8% of samples and for more samples than following the c-
$\mu$ rule or its extended version. Policy P2 is best for fewer samples (56.7%) when both 
$\mu _1$ and 
$\mu _2$ are low. In this case, P2 is again better for more samples than the c-
$\mu$ rule or its extended version. Meanwhile, policy P2 is best for only 41.6% of samples and for fewer samples than the extended c-
$\mu$ rule (59.2%) when 
$\mu _1$ is high and 
$\mu _2$ is low. When 
$\mu _1$ and 
$\mu _2$ are both high, policy P2 is once again best for a majority of samples (71.1%) but still best for fewer samples than the extended c-
$\mu$ rule (78.4%). Importantly, the extended c-
$\mu$ rule performs better than P2 for lower approximate loads under P2 (averaged over the relevant samples).
Figure 1 depicts an example when the extended c-
$\mu$ rule helps guide whether to use P2 or P1. Parameters 
$\mu _1$ and 
$\mu _2$ were systematically varied. Other parameters were fixed at values specified earlier; the coefficient of variation (cv) was 
$1.4$. Policy P2 is best for most 
$\mu _1$ and 
$\mu _2$ values except when the extended c-
$\mu$ inequality favors P1 or is close to favoring P1. Policy P1 is best for remaining 
$\mu _1$ and 
$\mu _2$ values except in one case when policy Inc is best. Other figures are presented in Appendices A.1 and A.2.
Figure 1. Average cost comparison for the multi-server model when the cv is fixed at 
$1.4$ and service rates 
$\mu _1$ and 
$\mu _2$ are varied.
Samples were then stratified by service rates and the cv for service and abandonment times (Table 3). For each pair of service rates 
$\mu _1$ and 
$\mu _2$, the policy P2 is the best policy for more samples when the cv is high (1.4) versus low (0.6). This improvement in P2 comes at the expense of P1, wherein P1 is the best policy for an increasing number of samples when the cv increases from 0.6 to 1.4. Even with the additional stratification on cv, the extended c-
$\mu$ rule still performs better than P2 for lower approximate loads at P2.
Table 3. Percent samples for which given policy yields lowest average costs as a function of service rates (
$\mu _1$ and 
$\mu _2$), approximate load under P2, and coefficient of variation (cv).
With P2 performing best in most samples, we wanted to characterize parameter values when a policy other than P2 performs well. We start with P1. Among the 128 cases of parameters in our factorial design (
$\mu _1$, 
$\mu _2$, 
$\beta _1$, 
$\beta _2$, 
$p$, and cv), P1 is best for a majority of samples for 20 of these cases compared to 88 for P2 (Table A.1 in Appendix). As one would expect, all 20 cases have a service rate 
$\mu _1$ faster or equal to 
$\mu _2$. Moreover, at least 16 of these cases have at least a high value 
$\mu _1$, low value of 
$\mu _2$; or low joining probability 
$p$. For 15 of these 20 cases, the extended c-
$\mu$ rule performs better than P1; these 15 cases coincide exactly with an approximate load under P2 lower than its mean of 2.2. In addition to a larger load, all of the remaining five cases are accompanied with low 
$\mu _2$, low 
$\beta _1$, high 
$\beta _2$, and low cv; neither 
$\lambda _2$, 
$\mu _1$, or 
$p$ took a consistent value across these cases.
Table 4 identifies parameter values (
$\mu _1$, 
$\mu _2$, 
$\beta _1$, 
$\beta _2$, 
$p$, and cv) when (non-priority) threshold policies are best for a majority of samples. We make three observations. First, there is no set of parameters, among the 128 considered, in which policy Exh performs best for a majority of samples. Second, the abandonment rate 
$\beta _1$ at phase 1 is low for every parameter case when a non-strict priority rule is optimal; neither 
$\lambda _2$, 
$\mu _1$, 
$\mu _2$, 
$\beta _2$, 
$p$, or cv took a consistent value across these cases. Third, the approximate load under P2 was larger than its average of 2.2 in all these cases when a (non-priority) threshold policy is optimal.
Table 4. Percent samples that policy yields lowest average costs in parameter cases when (non-priority) threshold policies are best for a majority of samples.
We summarize insights gained from the numerical study as follows. Using the extended c-
$\mu$ rule should guide when to use P2 or P1 policies, provided the approximate load under P2 is low. When the approximate load under P2 is high, then P2 is likely the best policy, but not always. The poorer relative performance of P2 and of the extended c-
$\mu$ rule occurs when customers abandon from phase 1 at a slow rate 
$\beta _1$, which can lead to long queues at phase 1, or when the variability in abandonment and service times is smaller than the mean time. Non-strict priority rules such as Inc, P1(5), and P2(5) (though not Exh) may even perform better than P1 or P2 in these situations.
6. Conclusion
In this paper, we allow for abandonments to a stochastic scheduling model consisting of a two-class, two stage tandem service system where we have generalized the models considered in Down et al. [Reference Down, Koole and Lewis14] and Zayas-Cabán et al. [Reference Zayas-Cabán, Xie, Green and Lewis49] by allowing customers to join phase 2 service after completing phase 1 service and by considering the performance criterion of minimizing holding costs per customer per unit time and lump-sum abandonment costs. Our main results are conditions for the optimality of static priority rules for the single-server model. Abandonments lead to technical challenges since the abandonment rate is not bounded, and uniformization is not possible. This means the standard induction arguments cannot be readily used. Furthermore, interchange arguments are difficult to apply since customers may abandon in between services. We use the CTMDP framework to analyze this decision-making scenario, and in particular, use the continuous-time optimality equations, and a sample path argument to show the results.
Because the optimal policy remains elusive outside the conditions provided in Theorems 4.5 and 4.6 and because our model relies on assumptions that may be unrealistic for certain systems of interest, we compare the long-run average costs between priority rules, based on the direct analog of the classic c-
$\mu$ rule when we add abandonments, with more complicated threshold policies in a discrete-event simulation study. We focus on a multi-server model where preemption and abandonments during service are not allowed, and when service and abandonment times are no longer exponentially distributed.
There are several venues for future research. Characterizing the optimal policy in general is of clear interest. For example, one way to show the optimality of switching curve policies is to show convexity and submodularity of the value function for the discounted cost problem and of the relative value function for the long-run average cost problem. We have attempted to prove these results in general, but up to this point have been unable to do so. Extending versions of Theorems 4.5 and 4.6 to when abandonments during service are not allowed is also of interest. In the latter case, the allocation decision now impacts the abandonment rates and holding costs per unit time, which suggest stronger conditions than those in Theorems 4.5 and 4.6 are needed to prove the optimality of static priority rules. Another extension is one in which there multiple classes in each phase of service, but one class always having priority over the other. While priorities reduce the complexity of such a proof using inductive or sample path arguments, the number of cases to consider remains high. Lastly, another venue is the study of asymptotically optimal policies akin to the analysis in Larrañaga et al. [Reference Larrañaga, Ayesta and Verloop31] or in James et al. [Reference James, Glazebrook and Lin19] for parallel queues with abandonments.
Supplementary material
To view supplementary material for this article, please visit https://doi.org/10.1017/S0269964822000213.
A. Appendix
Parameter cases when P1 performs best
A.1. A.1. Additional simulations when ${\rm cv} = 1.6$
Parameters were systematically varied to examine further when to use various policies, keeping other parameters fixed. The coefficient of variation, cv, was fixed at 1.6. Recall that the main text reports the case when 
$\mu _1$ and 
$\mu _2$ are systematically varied. Varying holding cost rates 
$h_1$ and 
$h_2$ also shows that P2 performs well when the extended c-
$\mu$ inequality favors P2 (Figure A.1). Policy P1 performs well when this inequality favors P1. In addition, we again find cases when a non-priority rule, in this case policy Exh, outperforms both priority rules, but the gap between Exh and P2 is small in these cases (
$\sim$0.1%).
Figure A.1. Average cost comparison for the multi-server model when cv is 1.6 and holding cost rates 
$h_1$ and 
$h_2$ are varied.
Varying abandonment rates 
$\beta _1$ and 
$\beta _2$ (Figure A.2) yields situations when policy P2 can perform better than P1 even when the extended c-
$\mu$ inequality favors P1. In addition, we find several cases when the threshold policy P1(5) performs better than the other heuristic policies, albeit it is close to P2 (
$\sim$0.3% away). By contrast, the extended c-
$\mu$ inequality provides perfect guidance when varying the arrival rate 
$\lambda _2$ to phase 2 or the joining probability 
$p$ (Figure A.3). That is, for these values of 
$p$ and 
$\lambda _2$, policy P2 performs better than P1 when the extended c-
$\mu$ inequality favors P2, whereas policy P1 performs better than P2 when the extended c-
$\mu$ inequality favors P1.
Figure A.2. Average cost comparison for the multi-server model when cv is 1.6 and abandonment rates 
$\beta _1$ and 
$\beta _2$ are varied.
Figure A.3. Average cost comparison for the multi-server model when cv is 1.6 and the joining probability 
$p$ and arrival rate 
$\lambda _2$ are varied.
A.2. A.2. Additional simulations when ${\rm cv} = 0.4$
Parameters were again systematically varied as before except the coefficient of variation, cv, is set to 0.4 as opposed to 1.6. Here, policy P2 performs best for all choices of 
$\mu _1$ and 
$\mu _2$, but its improvement over P1 is small (<5%) when the extended c-
$\mu$ inequality is violated. Thus, the extended c-
$\mu$ inequality can still help guide which priority rule performs well when varying service rates 
$\mu _1$ and 
$\mu _2$, but may not always yield the best policy (Figure A.4).
Figure A.4. Average cost comparison for the multi-server model when the cv is 0.4 and service rates 
$\mu _1$ and 
$\mu _2$ are varied.
Meanwhile, varying holding cost rates 
$h_1$ and 
$h_2$ (Figure A.5) reinforces the use of the extended c-
$\mu$ inequality. Policy P2 performs well when this inequality is satisfied and policy P2 performs well when this inequality is not satisfied. In addition, we again find cases when a threshold policy, that is, P1(5), that outperforms both priority rules, but the gap between this threshold policy and P2 is small in these cases (
$\sim$0–1%).
Figure A.5. Average cost comparison for the multi-server model when the cv is 0.4, and holding cost rates 
$h_1$ and 
$h_2$ are varied.
Varying abandonment rates 
$\beta _1$ and 
$\beta _2$ (Figure A.6) yields situations when policy P2 can perform better than P1 even when the extended c-
$\mu$ inequality is violated. This occurs when the abandonment rate 
$\beta _2$ is low, reinforcing what we found in Scenario 1 (and in the single-server model): that neglecting phase 2 when there are few abandonments at 2 can yield poor performance. In addition, we find several cases when the threshold policy P1(5) performs better than the other heuristic policies, albeit it is close to P2 (
$\leq$1% away).
Figure A.6. Average cost comparison for the multi-server model when the cv is 0.4, and abandonment rates 
$\beta _1$ and 
$\beta _2$ are varied.
Last, we varied the joining probability 
$p$ and the arrival rate 
$\lambda _2$ (Figure A.7). Similar to what we observed when we varied 
$\mu _1$ and 
$\mu _2$, policy P2 performs better than policy P1 in all choices of 
$p$ and 
$\lambda _2$, and best in most in most choices, but its improvement over P1 is small (<1%) when the extended c-
$\mu$ inequality is violated. Thus, the extended c-
$\mu$ inequality can still help guide which priority rule performs well when varying service rates 
$\mu _1$ and 
$\mu _2$, but may not always yield the best policy.
Figure A.7. Average cost comparison for the multi-server model when the cv is 0.4 and joining probability 
$p$ and arrival rate 
$\lambda _2$ are varied.
Table A.1. Percent samples that policy yields lowest average costs in parameter cases when P1 is best for a majority of samples.
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