Appendix A: Introduction to Queueing Theory

Transcription

1 Appendix A: Introduction to Queueing Theory Queueing theory is an advanced mathematical modeling technique that can estimate waiting times. Imagine customers who wait in a checkout line at a grocery store. This waiting line is referred to as a queue. Customers who finish their shopping will wait in the queue until a checkout clerk is available to scan the items in their carts. Since customers arrive randomly and the number of items they have can vary substantially, the delays they encounter are highly variable and depend upon the number of clerks who are working and how fast they can work. A queueing model can be used to translate the arrival pattern and processing times to estimate important system performance measures, e.g., customer average waiting times and the likelihood of a random customer encountering zero delays, for any number of clerks. In our study, patient appointment calls play the role of customers in a store, patient appointment schedule serves as the queue, and providers act as store clerks who are needed to provide the required service. Thus we can use queueing models to evaluate the productivity and cost-efficiencies of different practice models by allowing the estimation of average AWTs. Queueing analysis is one of the most practical and effective tools for understanding and aiding decisionmaking in managing critical resources (Green 2006). For recent studies using queueing theory in health services research, please refer to Green (2002) and Green and Nguyen (2001). For a general technical reference, please refer to Gross and Harris (1985).

2 Appendix B: Analysis of the Models For model i=i, II and III, we denote its productivity under the exponential and deterministic service time distributions by N M i ( M stands for Markovian) and N D i, respectively. Dividing the total annual staffing costs of model i by its productivity, we obtain its annual costs per patient, denoted by c M i and c D i, respectively, under the two service time distributions. The cost-efficiency of Model i is then measured by these costs. Model I If the service time follows an exponential distribution, then the model becomes an M/M/1 queueing model, the simplest and most classical model in queueing theory (Gross, and Harris 1985). It follows from a standard Markov Chain analysis of such models that /. Given that, we obtain that / 1. Hence 1. If the services times are deterministic, numerical procedure to compute waiting times are available. The readers can refer to any standard queueing textbook for such methods, e.g., Gross and Harris (1985). Model II Model II is a special queueing network called Jackson Network (Gross, and Harris 1985) if service times follow exponential distributions. In this case, the NP queue and the PCP queue behave as if two independent M/M/1 queues with arrival rates and, respectively, where 1. It follows that the average waiting time of all patients can be calculated as if no supervision is required (replace by 0.5 or 1 when lower or high level of supervision is needed, respectively). Setting, we obtain, which is equivalent to the following quadratic function for :.

3 Solving it for, we obtain, where, and. Since the stability condition requires that 1 and 1, we have. Recall that and. We obtain 2 4 2, and thus 2 4. If service times are deterministic, model II does not have a closed-form solution but queueing network approximation algorithms are available. See, e.g., Gross and Harris (1985). Model III We model the system as a two-server queueing system where patients will be served by the NP or the PCP whoever becomes available. If a patient arrives at the system and both providers are free, the patient will be served by the PCP. Since the PCP has shorter consultation times, this setup provides better model performances. Consider the case where the service times follow exponential distributions. Let for 0,2,3,4, if there are totally patients waiting in both the NP queue and the PCP queue (including those in service) at time. Let 1,1 if there is only one patient in the system and being served by the NP at time, and let 1,2 if there is only one patient in the system and being served by the PCP at time. Thus the system state of Model III at time 0 can be described by, which evolves over time as a Continuous Time Markov Chain (CTMC) with state space Ω 0, 1,1, 1,2, 2,3,4. Let lim for Ω. The balance equations for, 0 can be written as follows.

4 ,,,, 2. Solving the above linear equation system, we obtain,,, and for 3, where,, Using the normalization condition that Ω,. 1, one can obtain for Ω, and in particular, It follows from Little s Law that Setting, we can calculate. Recall that and, we obtain. When the service times are deterministic, it is difficult to study the model analytically and hence we resort to a discrete-event simulation study (Law, and Kelton 1991). In the study, we simulate the operation of the clinic as a chronological sequence of events, which include arrival of patients, start of consultations, and completion of consultations (i.e., departure of patients). At any single point of time, only one event can occur. The timing of events is determined by relevant probability distributions associated with those

5 events. For example, time between two consecutive patient arrivals is drawn from an exponential distribution parameterized by patient arrival rate, since patient arrival is assumed to follow a Poisson Process. The choice of model parameters is discussed in the main text. The sequence of events, i.e., which event to occur next, relies on the timing of all events that could happen next. For example, the event following a patient arrival could be either another patient arrival or completion of a consultation, depending on which one would occur first. The simulation program is developed using Matlab software (The MathWorks, Inc., Natick, MA). By simulating the system and collecting relevant statistics over time, we can generate estimates for the mean and standard deviation of the AWT for a given panel size. We can then identify the largest panel size (i.e., productivity) under which the mean of average AWT equals the threshold T specified in the AWT requirement, i.e., 1 day in our study. Compared to analytical solutions which typically offer a quick and exact procedure to determine the relationship between system inputs and outputs, simulation modeling and analysis usually take a much longer time (to develop and run) and the outputs are usually random variables (since they are usually based on random inputs). Therefore, it may be hard to determine whether an observation is a result of system interrelations or randomness (Banks 1999). However, simulation allows consideration of more realistic/complex systems which would otherwise have to be simplified in order to obtain analytical solutions. Readers are referred to Law and Kelton (1991) and Fishman (2001) for more detailed introduction of discrete-event simulation.

6 Appendix C: Numerical Results of All Tested Cases Table 1. Results when NP service rate µ 1 =12 patients/day non-markovian Model Productivity Average Cost 1 Cost Ratio 2 Markovian Model Productivity Average Cost 1 Model I - Solo-physician 2440 $ % 2381 $ % Model II - Supervision Model (no supervision) Referral rate Cost Ratio 2 Substitution ratio 0% 3064 $ % 3004 $ % 20% 5% 3025 $ % 2965 $ % 10% 2988 $ % 2927 $ % 15% 2951 $ % 2891 $ % 20% 2915 $ % 2855 $ % 25% 2880 $ % 2820 $ % Substitution ratio 0% 3509 $ % 3444 $ % 30% 5% 3434 $ % 3369 $ % 10% 3362 $ % 3298 $ % 15% 3294 $ % 3230 $ % 20% 3228 $ % 3164 $ % 25% 3164 $ % 3101 $ % Substitution ratio 0% 3856 $ % 3747 $ % 38% 5% 3807 $ % 3694 $ % 10% 3718 $ % 3617 $ % 15% 3620 $ % 3530 $ % 20% 3523 $ % 3440 $ % 25% 3430 $ % 3351 $ % Substitution ratio 0% 3681 $ % 3602 $ % 40% 5% 3678 $ % 3590 $ % 10% 3669 $ % 3570 $ % 15% 3646 $ % 3533 $ % 20% 3584 $ % 3473 $ % 25% 3495 $ % 3397 $ % Substitution ratio 0% 2938 $ % 2876 $ % 50% 5% 2938 $ % 2875 $ % 10% 2938 $ % 2874 $ % 15% 2937 $ % 2873 $ % 20% 2937 $ % 2872 $ % 25% 2937 $ % 2870 $ % Substitution ratio 0% 2439 $ % 2380 $ % 60% 5% 2439 $ % 2379 $ %

7 Model II Supervision Model (Low level of supervision) 10% 2439 $ % 2379 $ % 15% 2439 $ % 2378 $ % 20% 2439 $ % 2378 $ % 25% 2439 $ % 2377 $ % Referral rate substitution ratio p 0% 2986 $ % 2926 $ % 20% 5% 2948 $ % 2888 $ % 10% 2912 $ % 2851 $ % 15% 2876 $ % 2816 $ % 20% 2841 $ % 2781 $ % 25% 2807 $ % 2747 $ % Substitution ratio 0% 3420 $ % 3355 $ % 30% 5% 3347 $ % 3283 $ % 10% 3277 $ % 3213 $ % 15% 3211 $ % 3146 $ % 20% 3146 $ % 3082 $ % 25% 3084 $ % 3020 $ % Substitution ratio 0% 3818 $ % 3704 $ % 38% 5% 3731 $ % 3629 $ % 10% 3630 $ % 3540 $ % 15% 3531 $ % 3447 $ % 20% 3435 $ % 3356 $ % 25% 3344 $ % 3268 $ % Substitution ratio 0% 3679 $ % 3593 $ % 40% 5% 3672 $ % 3575 $ % 10% 3652 $ % 3541 $ % 15% 3596 $ % 3484 $ % 20% 3506 $ % 3407 $ % 25% 3411 $ % 3321 $ % Substitution ratio 0% 2938 $ % 2876 $ % 50% 5% 2938 $ % 2875 $ % 10% 2938 $ % 2874 $ % 15% 2937 $ % 2872 $ % 20% 2937 $ % 2871 $ % 25% 2937 $ % 2869 $ % Substitution ratio 0% 2439 $ % 2379 $ % 60% 5% 2439 $ % 2379 $ % 10% 2439 $ % 2379 $ % 15% 2439 $ % 2378 $ % 20% 2439 $ % 2378 $ %

8 Model II Supervision Model (High level of supervision) 25% 2439 $ % 2377 $ % Referral rate Substitution ratio 0% 2908 $ % 2848 $ % 20% 5% 2871 $ % 2811 $ % 10% 2835 $ % 2775 $ % 15% 2801 $ % 2741 $ % 20% 2767 $ % 2707 $ % 25% 2733 $ % 2673 $ % Substitution ratio 0% 3330 $ % 3267 $ % 30% 5% 3259 $ % 3196 $ % 10% 3191 $ % 3128 $ % 15% 3126 $ % 3063 $ % 20% 3063 $ % 3000 $ % 25% 3003 $ % 2940 $ % Substitution ratio 0% 3745 $ % 3640 $ % 38% 5% 3642 $ % 3550 $ % 10% 3539 $ % 3456 $ % 15% 3441 $ % 3362 $ % 20% 3347 $ % 3271 $ % 25% 3257 $ % 3184 $ % Substitution ratio 0% 3674 $ % 3580 $ % 40% 5% 3658 $ % 3549 $ % 10% 3607 $ % 3495 $ % 15% 3518 $ % 3418 $ % 20% 3421 $ % 3331 $ % 25% 3325 $ % 3241 $ % Substitution ratio 0% 2938 $ % 2875 $ % 50% 5% 2938 $ % 2874 $ % 10% 2937 $ % 2873 $ % 15% 2937 $ % 2871 $ % 20% 2937 $ % 2870 $ % 25% 2936 $ % 2867 $ % Substitution ratio 0% 2439 $ % 2379 $ % 60% 5% 2439 $ % 2379 $ % Model III Shared-panel model 10% 2439 $ % 2379 $ % 15% 2439 $ % 2378 $ % 20% 2439 $ % 2377 $ % 25% 2438 $ % 2377 $ % 3942 $ % 3881 $ %

9 Table 2. Results when NP service rate µ 1 =15 patients/day non-markovian Model Productivity Average Cost 1 Cost Ratio 2 Markovian Model Productivity Average Cost 1 Model I - Solo-physician 2440 $ % 2381 $ % Model II - Supervision Model (no supervision) Referral rate Cost Ratio 2 Substitution ratio 0% 3064 $ % 3004 $ % 20% 5% 3025 $ % 2966 $ % 10% 2988 $ % 2928 $ % 15% 2951 $ % 2891 $ % 20% 2915 $ % 2856 $ % 25% 2880 $ % 2821 $ % Substitution ratio 0% 3510 $ % 3448 $ % 30% 5% 3435 $ % 3373 $ % 10% 3363 $ % 3301 $ % 15% 3294 $ % 3233 $ % 20% 3228 $ % 3167 $ % 25% 3,165 $ % 3103 $ % Substitution ratio 0% 4099 $ % 4023 $ % 40% 5% 3966 $ % 3893 $ % 10% 3842 $ % 3771 $ % 15% 3724 $ % 3655 $ % 20% 3614 $ % 3545 $ % 25% 3509 $ % 3441 $ % Substitution ratio 0% 4251 $ % 4136 $ % 43% 5% 4143 $ % 4040 $ % 10% 4006 $ % 3917 $ % 15% 3872 $ % 3791 $ % 20% 3745 $ % 3669 $ % 25% 3626 $ % 3552 $ % Substitution ratio 0% 3687 $ % 3621 $ % 50% 5% 3686 $ % 3618 $ % 10% 3685 $ % 3615 $ % 15% 3684 $ % 3610 $ % 20% 3682 $ % 3602 $ % 25% 3677 $ % 3586 $ % Substitution ratio 0% 3064 $ % 3003 $ % 60% 5% 3064 $ % 3002 $ % 10% 3063 $ % 3001 $ % 15% 3063 $ % 3001 $ %

10 Model II Supervision Model (Low level of supervision) 20% 3063 $ % 3000 $ % 25% 3063 $ % 2998 $ % Referral rate substitution ratio p 0% 2986 $ % 2926 $ % 20% 5% 2948 $ % 2889 $ % 10% 2912 $ % 2852 $ % 15% 2876 $ % 2816 $ % 20% 2841 $ % 2781 $ % 25% 2807 $ % 2747 $ % Substitution ratio 0% 3421 $ % 3359 $ % 30% 5% 3348 $ % 3286 $ % 10% 3278 $ % 3216 $ % 15% 3210 $ % 3149 $ % 20% 3146 $ % 3085 $ % 25% 3084 $ % 3023 $ % Substitution ratio 0% 3996 $ % 3923 $ % 40% 5% 3866 $ % 3795 $ % 10% 3744 $ % 3675 $ % 15% 3630 $ % 3562 $ % 20% 3522 $ % 3454 $ % 25% 3420 $ % 3353 $ % Substitution ratio 0% 4184 $ % 4075 $ % 43% 5% 4046 $ % 3954 $ % 10% 3907 $ % 3825 $ % 15% 3775 $ % 3698 $ % 20% 3651 $ % 3577 $ % 25% 3535 $ % 3463 $ % Substitution ratio 0% 3687 $ % 3620 $ % 50% 5% 3686 $ % 3617 $ % 10% 3685 $ % 3612 $ % 15% 3683 $ % 3606 $ % 20% 3679 $ % 3593 $ % 25% 3670 $ % 3568 $ % Substitution ratio 0% 3064 $ % 3002 $ % 60% 5% 3063 $ % 3002 $ % 10% 3063 $ % 3001 $ % 15% 3063 $ % 3000 $ % 20% 3063 $ % 2999 $ % 25% 3062 $ % 2998 $ %

11 Model II Supervision Model (High level of supervision) Referral rate Substitution ratio 0% 2908 $ % 2848 $ % 20% 5% 2871 $ % 2812 $ % 10% 2836 $ % 2776 $ % 15% 2801 $ % 2741 $ % 20% 2767 $ % 2707 $ % 25% 2733 $ % 2674 $ % Substitution ratio 0% 3331 $ % 3270 $ % 30% 5% 3260 $ % 3199 $ % 10% 3192 $ % 3131 $ % 15% 3127 $ % 3065 $ % 20% 3064 $ % 3003 $ % 25% 3004 $ % 2942 $ % Substitution ratio 0% 3893 $ % 3822 $ % 40% 5% 3766 $ % 3697 $ % 10% 3647 $ % 3579 $ % 15% 3535 $ % 3468 $ % 20% 3430 $ % 3364 $ % 25% 3331 $ % 3265 $ % Substitution ratio 0% 4089 $ % 3993 $ % 43% 5% 3945 $ % 3861 $ % 10% 3807 $ % 3730 $ % 15% 3678 $ % 3604 $ % 20% 3557 $ % 3485 $ % 25% 3443 $ % 3372 $ % Substitution ratio 0% 3686 $ % 3618 $ % 50% 5% 3685 $ % 3615 $ % 10% 3684 $ % 3609 $ % 15% 3681 $ % 3599 $ % 20% 3674 $ % 3580 $ % 25% 3652 $ % 3538 $ % Substitution ratio 0% 3064 $ % 3002 $ % 60% 5% 3063 $ % 3002 $ % Model III Shared-panel model 10% 3063 $ % 3001 $ % 15% 3063 $ % 3000 $ % 20% 3063 $ % 2998 $ % 25% 3062 $ % 2997 $ % 4317 $ % 4256 $ %

12 Table 3. Results when NP service rate µ 1 =18 patients/day non-markovian Model Productivity Average Cost 1 Cost Ratio 2 Markovian Model Productivity Average Cost 1 Model I - Solo-physician 2440 $ % 2381 $ % Model II - Supervision Model (no supervision) Referral rate Cost Ratio 2 Substitution ratio 0% 3064 $ % 3004 $ % 20% 5% 3026 $ % 2966 $ % 10% 2988 $ % 2928 $ % 15% 2951 $ % 2892 $ % 20% 2915 $ % 2856 $ % 25% 2880 $ % 2821 $ % Substitution ratio 0% 3510 $ % 3449 $ % 30% 5% 3435 $ % 3374 $ % 10% 3363 $ % 3303 $ % 15% 3295 $ % 3234 $ % 20% 3228 $ % 3168 $ % 25% 3165 $ % 3104 $ % Substitution ratio 0% 4104 $ % 4038 $ % 40% 5% 3970 $ % 3905 $ % 10% 3844 $ % 3780 $ % 15% 3726 $ % 3662 $ % 20% 3615 $ % 3551 $ % 25% 3510 $ % 3446 $ % Substitution ratio 0% 4621 $ % 4510 $ % 47% 5% 4443 $ % 4353 $ % 10% 4265 $ % 4185 $ % 15% 4098 $ % 4023 $ % 20% 3943 $ % 3871 $ % 25% 3799 $ % 3728 $ % Substitution ratio 0% 4432 $ % 4355 $ % 50% 5% 4426 $ % 4336 $ % 10% 4398 $ % 4284 $ % 15% 4268 $ % 4165 $ % 20% 4098 $ % 4011 $ % 25% 3935 $ % 3856 $ % Substitution ratio 0% 3688 $ % 3626 $ % 60% 5% 3688 $ % 3625 $ % 10% 3688 $ % 3623 $ % 15% 3687 $ % 3622 $ %

13 Model II Supervision Model (Low level of supervision) 20% 3686 $ % 3619 $ % 25% 3686 $ % 3615 $ % Referral rate substitution ratio p 0% 2986 $ % 2926 $ % 20% 5% 2948 $ % 2889 $ % 10% 2912 $ % 2852 $ % 15% 2876 $ % 2816 $ % 20% 2841 $ % 2782 $ % 25% 2807 $ % 2747 $ % Substitution ratio 0% 3421 $ % 3360 $ % 30% 5% 3348 $ % 3287 $ % 10% 3278 $ % 3217 $ % 15% 3211 $ % 3150 $ % 20% 3146 $ % 3086 $ % 25% 3084 $ % 3024 $ % Substitution ratio 0% 4000 $ % 3935 $ % 40% 5% 3869 $ % 3805 $ % 10% 3746 $ % 3682 $ % 15% 3631 $ % 3568 $ % 20% 3523 $ % 3459 $ % 25% 3421 $ % 3358 $ % Substitution ratio 0% 4521 $ % 4423 $ % 47% 5% 4334 $ % 4252 $ % 10% 4158 $ % 4082 $ % 15% 3995 $ % 3922 $ % 20% 3843 $ % 3773 $ % 25% 3702 $ % 3633 $ % Substitution ratio 0% 4430 $ % 4347 $ % 50% 5% 4417 $ % 4315 $ % 10% 4337 $ % 4224 $ % 15% 4167 $ % 4075 $ % 20% 3996 $ % 3915 $ % 25% 3836 $ % 3760 $ % Substitution ratio 0% 3688 $ % 3625 $ % 60% 5% 3688 $ % 3624 $ % 10% 3687 $ % 3623 $ % 15% 3687 $ % 3621 $ % 20% 3686 $ % 3617 $ % 25% 3685 $ % 3612 $ %

14 Model II Supervision Model (High level of supervision) Referral rate Substitution ratio 0% 2908 $ % 2848 $ % 20% 5% 2871 $ % 2812 $ % 10% 2836 $ % 2776 $ % 15% 2801 $ % 2741 $ % 20% 2767 $ % 2707 $ % 25% 2733 $ % 2674 $ % Substitution ratio 0% 3332 $ % 3271 $ % 30% 5% 3260 $ % 3200 $ % 10% 3192 $ % 3132 $ % 15% 3127 $ % 3066 $ % 20% 3064 $ % 3004 $ % 25% 3004 $ % 2943 $ % Substitution ratio 0% 3895 $ % 3831 $ % 40% 5% 3768 $ % 3704 $ % 10% 3649 $ % 3585 $ % 15% 3537 $ % 3473 $ % 20% 3431 $ % 3368 $ % 25% 3332 $ % 3269 $ % Substitution ratio 0% 4409 $ % 4322 $ % 47% 5% 4223 $ % 4146 $ % 10% 4051 $ % 3978 $ % 15% 3891 $ % 3820 $ % 20% 3743 $ % 3674 $ % 25% 3606 $ % 3538 $ % Substitution ratio 0% 4426 $ % 4334 $ % 50% 5% 4390 $ % 4274 $ % 10% 4240 $ % 4142 $ % 15% 4062 $ % 3978 $ % 20% 3893 $ % 3816 $ % 25% 3737 $ % 3663 $ % Substitution ratio 0% 3688 $ % 3625 $ % 60% 5% 3688 $ % 3624 $ % Model III Shared-panel model 10% 3687 $ % 3622 $ % 15% 3687 $ % 3619 $ % 20% 3686 $ % 3616 $ % 25% 3684 $ % 3609 $ % 1 Average cost: average annual staffing cost per patient 2 Cost ratio: the ratio of the average costs of a model to that of Model I (solo-physician model) 4689 $ % 4630 $ %

15 Appendix D: Comparison of Markovian and non-markovian Models As expected, the productivity (and hence cost-efficiency) of a non-markovian model is larger than that of its Markovian counterpart with everything else being equal (see Appendix C). However, the percentage gap in the productivity (and cost-efficiency) estimated under these two service time assumptions is quite small (only 2~3% across all testing cases), indicating that system performances are not very sensitive to the variability in provider consultation times. For example, the productivity of Model II (assuming µ 1 =15, p=40%, r=0% and no PCP supervision) is estimated to be 4023 and 4099 under Markovian and non-markovian service time assumptions, respectively, and the percentage gap between them is less than 2%. In addition, the qualitative comparison results among different practice models are the same regardless of service time assumptions. For example, under both service time assumptions, Model II with µ 1 =15, p=30%, r=0% and no supervisions is more cost-efficient than Model I, and this comparative result reverses when r 10%. More importantly, the relative performance among these practice models is highly consistent across different service time assumptions. For all scenarios tested for Models II and III, the difference in cost ratios under the two service time assumptions is less than 1%, where cost ratio is defined as the ratio of the average annual cost of a model to that of Model I (our benchmark model). Considering that the two service time distributions we used represent two extremes of the variability in provider consultation times, the comparative results across Models I, II and III are likely to be insensitive to such variability. In summary, both performances within a practice model and comparative results across these models seem not sensitive to the variability in provider consultation times, implying that our results can be generalizable to different consultation time distributions.