A Markov chain model for analysis of physician workflow in primary care clinics

Abstract

This paper studies physician workflow management in primary care clinics using terminating Markov chain models. The physician workload is characterized by face-to-face encounters with patients and documentation of electronic health record (EHR) data. Three workflow management policies are considered: preemptive priority (stop ongoing documentation tasks if a new patient arrives); non-preemptive priority (finish ongoing documentation even if a new patient arrives); and batch documentation (start and finish documentation when the desired number of tasks is reached). Analytical formulas are derived to quantify the performance measures of three management policies, such as physician’s daily working time, patient’s waiting time, and documentation waiting time. A comparison of the results under three policies is carried out. Finally, a case study in a primary care clinic is carried out to illustrate model applicability. Such a work provides a quantitative tool for primary care physicians to design and manage their workflow to improve care quality.

Appendices

Appendix A: Transitions

In PEP model, the transitions occur based on the following conditions:

• (P1) If ${X_{p}^{k}} = {X_{d}^{k}} = 0$ and ${X_{c}^{k}} <N$, the physician is idle (no patient and documentation are in process or waiting); thus only Transition 1 can occur.

• (P2) If ${X_{p}^{k}} > 0$ and ${X_{p}^{k}} + {X_{d}^{k}} +{X_{c}^{k}} < N$, there are patients in the system so that the physician is meeting with them; thus both Transitions 1 and 2 can occur. But Transition 3 cannot occur because the physician is working on a patient.

• (P3) If ${X_{p}^{k}} > 0$ and ${X_{p}^{k}} + {X_{d}^{k}} + {X_{c}^{k}} =N$, there are patients in the system and no more patient will come; thus only Transition 2 can occur.

• (P4) If ${X_{p}^{k}} = 0$, ${X_{d}^{k}} > 0$ and ${X_{d}^{k}} + {X_{c}^{k}}< N$, no patient is in the system, but documentation is not finished and still more patients will come; thus both Transitions 1 and 3 can occur.

• (P5) If ${X_{p}^{k}} = 0$, ${X_{d}^{k}} > 0$ and ${X_{d}^{k}} + {X_{c}^{k}} =N$, no patient is in the system, documentation is not done yet but no more patient will come; thus only Transition 3 can occur.

In NPP model, the transitions occur based on the following rules:

• (N1) If ${X_{p}^{k}} = {X_{d}^{k}} = 0$ and ${X_{c}^{k}} < N$, the physician is idle; thus only Transition 1 can occur.

• (N2) If ${D_{d}^{k}} = 0$, ${X_{p}^{k}} > 0$ and ${X_{p}^{k}} + {X_{d}^{k}} + {X_{c}^{k}} < N$, there are patients in the system, and more will come, but no documentation is in process; thus both Transitions 1 and 2 can occur.

• (N3) If ${D_{d}^{k}} = 0$, ${X_{p}^{k}} > 0$ and ${X_{p}^{k}} + {X_{d}^{k}} + {X_{c}^{k}} = N$, there are patients in the system, but no patient will come and no documentation is in process; thus only Transition 2 can occur.

• (N4) If ${D_{d}^{k}} = 0$, ${X_{p}^{k}} = 0$, ${X_{d}^{k}} > 0$ and ${X_{d}^{k}} + {X_{c}^{k}} < N$, no patient is in the system but more will come, and documentation is in process; thus both Transitions 3 and 4 can occur.

• (N5) If ${D_{d}^{k}} = 0$, ${X_{p}^{k}} = 0$, ${X_{d}^{k}} > 0$ and ${X_{d}^{k}} + {X_{c}^{k}}= N$, all patients have been served and documentation is in process; thus only Transition 3 can occur.

• (N6) If ${D_{d}^{k}} = 1$ and ${X_{p}^{k}} + {X_{d}^{k}} + {X_{c}^{k}} < N$, a new patient comes while documentation is in process, and more patients will come; thus both Transitions 3 and 4 can occur.

• (N7) If ${D_{d}^{k}} = 1$ and ${X_{p}^{k}} + {X_{d}^{k}} + {X_{c}^{k}} = N$, the last patient arrives while documentation is in process; thus only Transition 3 can occur.

In BDC model, the transitions occur in the following scenarios:

• (B1) If $({X_{c}^{k}} \bmod M) = 0$, ${X_{p}^{k}} = 0$, ${X_{d}^{k}} < M$ and ${X_{p}^{k}} +{X_{d}^{k}} + {X_{c}^{k}} < N$, no patient is in the system, and previous documentation batch is finished but current one is not started since the batch is not filled, so the physician is idle; thus only Transition 1 can occur.

• (B2) If $({X_{c}^{k}} \bmod M) = 0$, ${X_{p}^{k}} > 0$, ${X_{d}^{k}} < M$ and ${X_{p}^{k}} +{X_{d}^{k}} + {X_{c}^{k}} < N$, there are patients in the system, and previous documentation batch is finished but current one is not started due to less number of tasks in the batch, so the physician is meeting with a patient; thus both Transitions 1 and 2 can occur.

• (B3) If $({X_{c}^{k}} \bmod M) = 0$, ${X_{d}^{k}} = M$ and ${X_{p}^{k}} + {X_{c}^{k}} <N-M$, the documentation batch size is reached so the physician will start working on the batch, and more patients will come; thus both Transitions 1 and 3 can occur.

• (B4) If $({X_{c}^{k}} \bmod M) = 0$, ${X_{d}^{k}} = M$ and ${X_{p}^{k}} + {X_{c}^{k}} =N-M$, the documentation batch size is reached so the physician will start working on the batch, but no patient will come; thus only Transition 3 can occur.

• (B5) If $({X_{c}^{k}} \bmod M) = 0$, ${X_{p}^{k}} > 0$, ${X_{d}^{k}} < M$ and ${X_{p}^{k}} +{X_{d}^{k}} + {X_{c}^{k}} = N$, there are patients in the system, and previous documentation batch is finished and current one is not started due to less number of tasks in the batch, so the physician is meeting with a patient, but no new patient will come; thus only Transition 2 can occur.

• (B6) If $({X_{c}^{k}} \bmod M) \neq 0$ and ${X_{p}^{k}} + {X_{d}^{k}} + {X_{c}^{k}} < N$, the current documentation batch is in process and more patients will come; thus both Transitions 1 and 3 can occur.

• (B7) If $({X_{c}^{k}} \bmod M) = 0$, ${X_{p}^{k}} = 0$ and ${X_{d}^{k}} + {X_{c}^{k}} = N$, all patients have been served, so the physician will work on documentation of the last batch no matter the batch size is not reached; thus only Transition 3 can occur.

• (B8) If $({X_{c}^{k}} \bmod M) \neq 0$ and ${X_{p}^{k}} + {X_{d}^{k}} + {X_{c}^{k}} = N$, the current documentation batch is in process and no new patient will come; thus only Transition 3 can occur.

For the transition rate, ∀l = 0, 1,…,N − 1; n = 0, 1,…,N − l − 1; m = 0, 1,…,N − l − n − 1, in the PEP model, we have the following transition rates corresponding to Transitions 1 to 3 described in (P1)-(P5):

$$ \begin{array}{@{}rcl@{}} \text{From P1,2,4:} &\quad& \eta((m,n,l),(m+1,n,l)) = \lambda, \\ \text{From P2,3:} &\quad& \eta((m+1,n,l),(m,n+1,l)) = \mu_{p}, \\ \text{From P4,5:} &\quad& \eta((0,n+1,l),(0,n,l+1)) = \mu_{d}. \end{array} $$

In the NPP model, it follows from Transitions 1 to 3 explained in (N1)–(N7) that

$$ \begin{array}{@{}rcl@{}} \text{From N1:} &\quad&{\kern-3.2pc} \eta((0,n,l,0),(1,n,l,0)) = \lambda, \quad \text{if } n = 0,\\ \text{From N4:} &\quad&{\kern-3.2pc} \eta((0,n,l,0),(1,n,l,1)) = \lambda, \quad \text{if } n > 0,\\ \text{From N2:} &\quad&{\kern-3.2pc} \eta((m,n,l,0),(m+1,n,l,0)) = \lambda, \\ &\qquad \qquad &{}\text{if } m > 0,\\ \text{From N6:} &\quad&{\kern-3.2pc} \eta((m,n,l,1),(m+1,n,l,1)) = \lambda, \\ &\qquad \qquad &{}\text{if } m > 0, n > 0,\\ \text{From N2,3:} &\quad&{\kern-3.2pc} \eta((m+1,n,l,0),(m,n+1,l,0)) = \mu_{p},\\ \text{From N4,5:} &\quad&{\kern-3.2pc} \eta((0,n+1,l,0),(0,n,l+1,0)) = \mu_{d},\\ \text{From N6,7:} &\quad&{\kern-3.2pc} \eta((m,n+1,l,1),(m,n,l+1,0)) = \mu_{d}, \\ &\qquad \qquad&{}\text{if } m > 0. \end{array} $$

In the BDC model, from Transitions 1 to 3 outlined in (B1)–(B8), we obtain,

$$ \begin{array}{@{}rcl@{}} \text{From B1-3,6:} &\quad& \eta((m,n,l),(m+1,n,l)) = \lambda,\\ \text{From B2,5:} &\quad& \eta((m+1,n,l),(m,n+1,l)) = \mu_{p},\\ &&\text{if } (l \bmod M) = 0 \text{ and } n < M,\\ \text{From B3,6:} &\quad& \eta((m,n,l),(m,n-1,l+1)) = \mu_{d},\\ &&\text{if } ((l+n) \bmod M) = 0 \text{ and } 1\leq n \leq M,\\ \text{From B4,7,8:} &\quad& \eta((0,n,l),(0,n-1,l+1)) = \mu_{d},\\ &&\text{if } l+n =N \text{ and } 1\leq n \leq M. \end{array} $$

Appendix B: Proofs

Proof Proof of Lemma 1

For a given number of completed documentation task l, the number of patients in the system, m, ranges from 0 to N − l. With given l and m, the number of unfinished documents is from 0 to N − l − m (i.e., N − l − m + 1 cases). Thus,

$$\sum\limits_{m=0}^{N-l} (N-l-m+1)= \frac{(N+2-l)(N+1-l)}{2}.$$

For l ∈ [0,N], we obtain

$$ \begin{array}{@{}rcl@{}} K^{PEP} &=& \sum\limits^{N}_{l=0} \frac{(N+2-l)(N+1-l)}{2}\\ &=& \frac{1}{6}(N+1)(N+2)(N+3). \end{array} $$

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Proof Proof of Lemma 2

The state space constraints for the number of patients in the system m, the number of unfinished documentation tasks n and the number of documentations tasks l in the NPP model are identical to those in the PEP model. Therefore, when ${D_{d}^{k}}=0$, the number of feasible states equals to K^PEP. When D_d = 1, m and n cannot be 0 due to the constraints of D_d. The number of states for m = 0 or n = 0 is

$$\frac{(N+1)(N+2)}{2} + \frac{(N+1)(N+2)}{2} - (N+1) = (N+1)^{2}.$$

Thus, we have

$$ \begin{array}{@{}rcl@{}} K^{NPP} &=& 2K^{PEP} - (N+1)^{2}\\ &=& \frac{1}{3}(N+1)(N^{2}+2N+3) \end{array} $$

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Proof of Lemma 3

For a given number of completed documentation tasks l, when l is a multiple of the batch size M, the number of unfinished documentation tasks n varies from 0 to $\min \limits (M,N-l)$. Furthermore, with a fixed n, the number of patients in the system m varies from 0 to N − n − l (i.e., N − n − l + 1 cases).

Let $A = \lfloor \frac {N}{M}\rfloor$ and B = N − AM. If l = Mk, where k is a non-negative integer, then

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{n=0}^{M} (N-n-Mk+1)\\ &&\qquad = \frac{(M+1)[2N+2-M(1+2k)]}{2}, \quad \text{if } k<A,\\ &&\sum\limits_{n=0}^{N-MA} (N-n-MA+1)= (B+1) + B + {\dots} + 1 \\ &&\qquad = \frac{(B+2)(B+1)}{2}, \quad \text{if } k=A. \end{array} $$

If l = Mk + j where j is a positive integer and j < M, then, n takes M − j so that n < M and ((n + l) mod M) = 0 due to state space constraints, and m varies from 0 to N − M(1 + k). Therefore, when $Mk<l<\min \limits (M(1+k),N)$, the number of feasible states is (M − 1)(N + 1 − M(1 + k)) if k < A, and B if k = A. Thus, we can obtain

$$ \begin{array}{@{}rcl@{}} K^{BDC} &=& \sum\limits^{A-1}_{k=0}\left[\frac{(M+1)(2N+2-M(1+2k))}{2}\right.\\ &&\left.\quad +(M-1)(N+1-M(1+k))\vphantom{\frac{1}{2}}\right]\\ &&\quad + \frac{(B+2)(B+1)}{2} + B\\ &= &\frac{MA(4N-M-2MA+5)+(B^{2}+5B+2)}{2}. \end{array} $$

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Appendix C: Algorithms for C V = 0

To evaluate the performance measures when CV = 0, calculation algorithms can be introduced for each model. In these algorithms, ta, ts, and td denote inter-arrival, service, documentation times, and N and M represent total number of patients and documentation batch size, respectively. Note that closed formulas for $T_{0}^{\text {PEP}}$, $T_{0}^{\text {NPP}}$, $T_{0}^{\text {BDC}}$ and $W_{0}^{\text {PEP}}$ can be obtained through the algorithm.