Comments on: What happens when you add a new teller? http://www.johndcook.com/blog/2008/10/21/what-happens-when-you-add-a-new-teller/ The blog of John D. Cook Sat, 11 Feb 2012 22:42:11 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: R.S.S. http://www.johndcook.com/blog/2008/10/21/what-happens-when-you-add-a-new-teller/comment-page-1/#comment-126458 R.S.S. Fri, 30 Dec 2011 17:12:18 +0000 http://www.johndcook.com/blog/?p=712#comment-126458 Queuing Theory is fun. Thank your U.S. Navy for a lot of work in the area. Queuing Theory is fun. Thank your U.S. Navy for a lot of work in the area.

]]> By: David Anderson http://www.johndcook.com/blog/2008/10/21/what-happens-when-you-add-a-new-teller/comment-page-1/#comment-126457 David Anderson Fri, 30 Dec 2011 17:11:32 +0000 http://www.johndcook.com/blog/?p=712#comment-126457 I'm totally on board with Exponential arrivals, but wouldn't M/G/1 but a more accurate representation of a bank queue? Service times are probably much closer to Normal than Exponential. I know M/M/1 is nice and clean, but probably not that accurate. And the long tail of exponential service times is what creates lots of the backlog and excess waiting times. I’m totally on board with Exponential arrivals, but wouldn’t M/G/1 but a more accurate representation of a bank queue? Service times are probably much closer to Normal than Exponential. I know M/M/1 is nice and clean, but probably not that accurate. And the long tail of exponential service times is what creates lots of the backlog and excess waiting times.

]]> By: Neil Gunther http://www.johndcook.com/blog/2008/10/21/what-happens-when-you-add-a-new-teller/comment-page-1/#comment-16954 Neil Gunther Tue, 05 May 2009 02:35:52 +0000 http://www.johndcook.com/blog/?p=712#comment-16954 I see that you discuss R elsewhere in your blog, so here is the PDQ-R (Pretty Damn Quick in R) code for the above Bank model. <code> # PDQ v5.0 # Created by NJG on Monday, May 4, 2009 library(pdq) arrival_rate=5.8 * 2 # cust/hr service_time=10.0/60 # 10 minutes Init("M/M/m Blog Example") #define the workflow CreateOpen("deposit", arrival_rate) SetWUnit("Cust") SetTUnit("Hour") #define an m-server queue (here m = 2) CreateMultiNode(2,"teller", CEN, FCFS) #define the service demand on the server SetDemand("teller", "deposit", service_time) Solve(CANON) Report() </code> I see that you discuss R elsewhere in your blog, so here is the PDQ-R (Pretty Damn Quick in R) code for the above Bank model.

# PDQ v5.0
# Created by NJG on Monday, May 4, 2009
library(pdq)
arrival_rate=5.8 * 2 # cust/hr
service_time=10.0/60 # 10 minutes
Init("M/M/m Blog Example")
#define the workflow
CreateOpen("deposit", arrival_rate)
SetWUnit("Cust")
SetTUnit("Hour")
#define an m-server queue (here m = 2)
CreateMultiNode(2,"teller", CEN, FCFS)
#define the service demand on the server
SetDemand("teller", "deposit", service_time)
Solve(CANON)
Report()

]]> By: Neil Gunther http://www.johndcook.com/blog/2008/10/21/what-happens-when-you-add-a-new-teller/comment-page-1/#comment-16944 Neil Gunther Mon, 04 May 2009 21:47:33 +0000 http://www.johndcook.com/blog/?p=712#comment-16944 This example is dramatic because the lone teller is already near saturation (arrival rate ~= service rate) and the queue is growing unbounded. So naturally, adding another teller will halve the utilization of each and reduce the queue size significantly. I used <a href="http://www.perfdynamics.com/Tools/PDQcode.html" rel="nofollow">PDQ-R</a> to check your calculations. See below, although the formatting might die in this little Reply box. The original waiting line has 28 people in steady state and indeed the residence time is 5 hrs. In the 2nd scenario, a customer only waits about 30% of the time and their mean waiting time is about 3 mins (0.0508 Hours), as you stated. A more interesting question is, what happens if we now double the traffic rate into the bank with 2 tellers? See PDQ report 3. Even though the tellers will return to being as busy as the original lone teller, the waiting time is only 2.3757 Hours; half that in scenario 1. Dual tellers really win! The great feature of 2 tellers or a dual-core processor, is that the customer (or thread) at the *head* of the waiting line gets serviced by the next available teller. The benefit over a single teller is due to the variation about the mean service period S. With an exp service time dsn, the variance is the same magnitude as the mean. So, if one customer is opening a new bank account (long service period), other customers (with shorter transactions) can be serviced *in parallel* by the other teller. New question: What happens if a 3rd teller is added? :) <code> ======================================= ****** PDQ Model Results ******* ======================================= 1. Single teller model. Node Sched Resource Workload Class Demand ---- ----- -------- -------- ----- ------ 1 MSQ teller deposit TRANS 0.1667 ****** RESOURCE Performance ******* Metric Resource Work Value Unit ------ -------- ---- ----- ---- Throughput teller deposit 5.8000 Cust/Hour Utilization teller deposit 96.6667 Percent Queue length teller deposit 29.0000 Cust Waiting line teller deposit 28.0333 Cust Waiting time teller deposit 4.8333 Hour Residence time teller deposit 5.0000 Hour 2. Add another teller. Node Sched Resource Workload Class Demand ---- ----- -------- -------- ----- ------ 2 MSQ teller deposit TRANS 0.1667 ****** RESOURCE Performance ******* Metric Resource Work Value Unit ------ -------- ---- ----- ---- Throughput teller deposit 5.8000 Cust/Hour Utilization teller deposit 48.3333 Percent Queue length teller deposit 1.2613 Cust Waiting line teller deposit 0.2947 Cust Waiting time teller deposit 0.0508 Hour Residence time teller deposit 0.2175 Hour 3. Double the customer traffic as well. Node Sched Resource Workload Class Demand ---- ----- -------- -------- ----- ------ 2 MSQ teller deposit TRANS 0.1667 ****** RESOURCE Performance ******* Metric Resource Work Value Unit ------ -------- ---- ----- ---- Throughput teller deposit 11.6000 Cust/Hour Utilization teller deposit 96.6667 Percent Queue length teller deposit 29.4915 Cust Waiting line teller deposit 27.5582 Cust Waiting time teller deposit 2.3757 Hour Residence time teller deposit 2.5424 Hour </code> This example is dramatic because the lone teller is already near saturation (arrival rate ~= service rate) and the queue is growing unbounded. So naturally, adding
another teller will halve the utilization of each and reduce the queue size significantly.

I used PDQ-R to check your calculations. See below, although the formatting might die in this little Reply box. The original waiting line has 28 people in steady state and indeed the residence time is 5 hrs. In the 2nd scenario, a customer only waits about 30% of the time and their mean waiting time is about 3 mins (0.0508 Hours), as you stated.

A more interesting question is, what happens if we now double the traffic rate into the bank with 2 tellers? See PDQ report 3. Even though the tellers will return to being as busy as the original lone teller, the waiting time is only 2.3757 Hours;
half that in scenario 1. Dual tellers really win!

The great feature of 2 tellers or a dual-core processor, is that the customer (or thread) at the *head* of the waiting line gets serviced by the next available teller.
The benefit over a single teller is due to the variation about the mean service period S. With an exp service time dsn, the variance is the same magnitude as the mean. So, if one customer is opening a new bank account (long service period), other customers (with shorter transactions) can be serviced *in parallel* by the other teller.

New question: What happens if a 3rd teller is added? :)


=======================================
****** PDQ Model Results *******
=======================================

1. Single teller model.

Node Sched Resource Workload Class Demand
---- ----- -------- -------- ----- ------
1 MSQ teller deposit TRANS 0.1667

****** RESOURCE Performance *******

Metric Resource Work Value Unit
------ -------- ---- ----- ----
Throughput teller deposit 5.8000 Cust/Hour
Utilization teller deposit 96.6667 Percent
Queue length teller deposit 29.0000 Cust
Waiting line teller deposit 28.0333 Cust
Waiting time teller deposit 4.8333 Hour
Residence time teller deposit 5.0000 Hour

2. Add another teller.

Node Sched Resource Workload Class Demand
---- ----- -------- -------- ----- ------
2 MSQ teller deposit TRANS 0.1667

****** RESOURCE Performance *******

Metric Resource Work Value Unit
------ -------- ---- ----- ----
Throughput teller deposit 5.8000 Cust/Hour
Utilization teller deposit 48.3333 Percent
Queue length teller deposit 1.2613 Cust
Waiting line teller deposit 0.2947 Cust
Waiting time teller deposit 0.0508 Hour
Residence time teller deposit 0.2175 Hour

3. Double the customer traffic as well.

Node Sched Resource Workload Class Demand
---- ----- -------- -------- ----- ------
2 MSQ teller deposit TRANS 0.1667

****** RESOURCE Performance *******

Metric Resource Work Value Unit
------ -------- ---- ----- ----
Throughput teller deposit 11.6000 Cust/Hour
Utilization teller deposit 96.6667 Percent
Queue length teller deposit 29.4915 Cust
Waiting line teller deposit 27.5582 Cust
Waiting time teller deposit 2.3757 Hour
Residence time teller deposit 2.5424 Hour

]]> By: John Venier http://www.johndcook.com/blog/2008/10/21/what-happens-when-you-add-a-new-teller/comment-page-1/#comment-8280 John Venier Wed, 22 Oct 2008 16:27:22 +0000 http://www.johndcook.com/blog/?p=712#comment-8280 Wow, that's amazing! I never would have guessed, but when you put it in terms of <em>slack</em> it makes sense. Wow, that’s amazing! I never would have guessed, but when you put it in terms of slack it makes sense.

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