Response Time Introduction Prof Christian Terwiesch Example Physician

Response Time Introduction Prof. Christian Terwiesch

Example Physician office - Patients arrive, on average, every five minutes - It takes ten minutes to serve a patient - Patients are willing to wait Þ What is the implied utilization of the barber shop? Þ How long will patients have to wait? Prof. Christian Terwiesch

Example Physician office - Patients arrive, on average, every five minutes - It takes four minutes to serve a patient - Patients are willing to wait Þ What is the utilization of the barber shop? Þ How long will patients have to wait? Prof. Christian Terwiesch

A Somewhat Odd Service Process Arrival Time Service Time 1 0 4 2 5 4 3 10 4 4 15 4 5 20 4 6 25 4 7 30 4 8 35 4 9 40 4 10 45 4 11 50 4 12 55 4 Patient 7: 00 7: 10 7: 20 Prof. Christian Terwiesch 7: 30 7: 40 7: 50 8: 00

A More Realistic Service Process Patient 1 Service Time 1 0 5 2 7 6 3 9 7 4 12 6 5 18 5 6 22 2 7 25 4 8 30 3 9 36 4 Patient 10 45 2 11 51 2 12 55 3 Patient 2 Patient 5 Patient 4 Patient 7 Patient 6 Patient 9 Patient 8 Patient 11 Patient 10 Patient 12 Time 7: 00 7: 10 7: 20 7: 30 7: 40 7: 50 3 2 Number of cases Arrival Time Patient 3 1 0 2 min. 3 min. 4 min. 5 min. Service times Prof. Christian Terwiesch 6 min. 7 min. 8: 00

Variability Leads to Waiting Time Patient 1 2 3 4 5 6 7 8 9 10 11 12 Arrival Time 0 7 9 12 18 22 25 30 36 45 51 55 Service Time 5 6 7 6 5 2 4 3 4 2 2 3 Service time Wait time 7: 00 7: 10 7: 20 7: 30 7: 40 7: 50 8: 00 5 4 3 2 Inventory (Patients at lab) 1 0 Prof. Christian Terwiesch

The Curse of Variability - Summary Variability hurts flow With buffers: we see waiting times even though there exists excess capacity Variability is BAD and it does not average itself out New models are needed to understand these effects Prof. Christian Terwiesch

Response Time Waiting time models: The need for excess capacity Prof. Christian Terwiesch

Modeling Variability in Flow Rate Minimum{Demand, Capacity} = Demand = 1/a Outflow No loss, waiting only This requires u<100% Outflow=Inflow Processing Buffer Inflow Demand process is “random” Processing p: average processing time Look at the inter-arrival times IA 1 IA 2 IA 3 IA 4 Same as “activity time” and “service time” Time a: average inter-arrival time CVa = CVp = St-Dev(processing times) Average(processing times) St-Dev(inter-arrival times) Average(inter-arrival times) Often Poisson distributed: CVa = 1 Constant hazard rate (no memory) Exponential inter-arrivals Can have many distributions: CVp depends strongly on standardization Often Beta or Log. Normal Difference between seasonality and variability Prof. Christian Terwiesch

The Waiting Time Formula Average flow time T Flow rate Inventory waiting Iq Inflow Entry to system Outflow Begin Service Time in queue Tq Increasing Variability Departure Service Time p Flow Time T=Tq+p Theoretical Flow Time Utilization Waiting Time Formula Variability factor Utilization factor Service time factor Prof. Christian Terwiesch 100%

Example: Walk-in Doc Newt Philly needs to get some medical advise. He knows that his Doc, Francoise, has a patient arrive every 30 minutes (with a standard deviation of 30 minutes). A typical consultation lasts 15 minutes (with a standard deviation of 15 minutes). The Doc has an open-access policy and does not offer appointments. If Newt walks into Francois’s practice at 10 am, when can he expect to leave the practice again? Prof. Christian Terwiesch

Summary Even though the utilization of a process might be less than 100%, it might still require long customer wait time Variability is the root cause for this effect As utilization approaches 100%, you will see a very steep increase in the wait time If you want fast service, you will have to hold excess capacity Prof. Christian Terwiesch

Response Time More on Waiting time models / Staffing to Demand Prof. Christian Terwiesch

Waiting Time Formula for Multiple, Parallel Resources Inventory in service Ip Inflow Inventory waiting Iq Outflow Flow rate Entry to system Begin Service Time in queue Tq Departure Service Time p Flow Time T=Tq+p Waiting Time Formula for Multiple (m) Servers Prof. Christian Terwiesch

Example: Online retailer Customers send emails to a help desk of an online retailer every 2 minutes, on average, and the standard deviation of the inter-arrival time is also 2 minutes. The online retailer has three employees answering emails. It takes on average 4 minutes to write a response email. The standard deviation of the service times is 2 minutes. Estimate the average customer wait before being served. Prof. Christian Terwiesch

Summary of Queuing Analysis Flow unit Server Utilization (Note: make sure <1) Inventory in service Ip Time related measures Inventory waiting Iq Outflow Inventory related measures (Flow rate=1/a) Entry to system Begin Service Waiting Time Tq Departure Service Time p Flow Time T=Tq+p Prof. Christian Terwiesch

Staffing Decision Customers send emails to a help desk of an online retailer every 2 minutes, on average, and the standard deviation of the inter-arrival time is also 2 minutes. The online retailer has three employees answering emails. It takes on average 4 minutes to write a response email. The standard deviation of the service times is 2 minutes. How many employees would we have to add to get the average wait time reduced to x minutes? Prof. Christian Terwiesch

What to Do With Seasonal Data Measure the true demand data Apply waiting model in each slice Slice the data by the hour (30 min, 15 min) Level the demand Assume demand is “stationary” within a slice Prof. Christian Terwiesch

Service Levels in Waiting Systems Fraction of customers who have to wait x seconds or less 1 0. 8 90% of calls had to wait 25 seconds or less Waiting times for those customers who do not get served immediately 0. 6 0. 4 Fraction of customers who get served without waiting at all 0. 2 0 0 50 100 150 200 Waiting time [seconds] • Target Wait Time (TWT) • Service Level = Probability{Waiting Time TWT} • Example: Big Call Center - starting point / diagnostic: 30% of calls answered within 20 seconds - target: 80% of calls answered within 20 seconds Prof. Christian Terwiesch

Response Time Capacity Pooling Prof. Christian Terwiesch

Managerial Responses to Variability: Pooling Independent Resources 2 x(m=1) Example: Processing time=4 minutes Inter-arrival time=5 minutes (at each server) m=1, Cva=CVp=1 ÞTq = Pooled Resources (m=2) Processing time=4 minutes Inter-arrival time=2. 5 minutes m=2, Cva=CVp=1 ÞTq = Prof. Christian Terwiesch

Managerial Responses to Variability: Pooling Waiting Time Tq 70. 00 m=1 60. 00 50. 00 40. 00 m=2 30. 00 20. 00 m=5 10. 00 m=10 60% 65% 70% 75% 80% 85% 90% 95% Utilization u Prof. Christian Terwiesch

Pooling: Shifting the Efficient Frontier Prof. Christian Terwiesch

Summary What is a good wait time? Fire truck or IRS? Prof. Christian Terwiesch

Limitations of Pooling Assumes flexibility Increases complexity of work-flow Can increase the variability of service time Interrupts the relationship with the customer / one-face-to-the-customer Group clinics Electricity grid / smart grid Flexible production plants Prof. Christian Terwiesch

The Three Enemies of Operations Additional costs due to variability in demand activity times Is associated with longer wait times and / or customer loss Requires process to hold excess capacity (idle time) Variability Waste Use of resources beyond what is needed to meet customer requirements • Not adding value to the product, but adding cost • Reducing the performance of the production system • 7 different types of waste Inflexibility Work Waste Value- Work Waste Valueadding Additional costs incurred because of supply demand mismatches • Waiting customers or • Waiting (idle capacity) Prof. Christian Terwiesch Customer demand Capacity

Response Time Scheduling / Access Prof. Christian Terwiesch

Managerial Responses to Variability: Priority Rules in Waiting Time Systems • Flow units are sequenced in the waiting area (triage step) • Provides an opportunity for us to move some units forwards and some backwards • First-Come-First-Serve - easy to implement - perceived fairness - lowest variance of waiting time • Sequence based on importance - emergency cases - identifying profitable flow units Prof. Christian Terwiesch

Managerial Responses to Variability: Priority Rules in Waiting Time Systems A Service times: A: 9 minutes B: 10 minutes C: 4 minutes D: 8 minutes C 9 min. 19 min. B D 4 min. 12 min. C 23 min. Total wait time: 9+19+23=51 min D A 21 min. Total wait time: 4+13+21=38 min • Shortest Processing Time Rule - Minimizes average waiting time - Problem of having “true” processing times Prof. Christian Terwiesch B

Appointments • Open Access • Appointment systems Prof. Christian Terwiesch

Response Time Redesign the Service Process Prof. Christian Terwiesch

Reasons for Long Response Times (And Potential Improvement Strategies) § Insufficient capacity on a permanent basis => Understand what keeps the capacity low § Demand fluctuation and temporal capacity shortfalls Unpredictable wait times => Extra capacity / Reduce variability in demand Predictable wait times => Staff to demand / Takt time § Long wait times because of low priority => Align priorities with customer value § Many steps in the process / poor internal process flow (often driven by handoffs and rework loops) => Redesign the service process http: //www. minyanville. com/businessmarkets/articles/drive-thrus-emissions-fast-food-mcdonalds/5/12/2010/id/28261 Prof. Christian Terwiesch

The Customer’s Perspective How much time does a patient spend on a primary care encounter? 20 minutes Driving Parking Check-in Vitals Waiting PCP Appt. Check out Labs Drive home Two types of wasted time: Auxiliary activities required to get to value add activities (result of process location / lay-out) Wait time (result of bottlenecks / insufficient capacity) Flow Time Efficiency (or %VAT) = Total value add time of a unit Total time a unit is in the process Prof. Christian Terwiesch

Process Mapping / Service Blue Prints Customer actions Walk into the branch / talk to agent Customer supplies more data Sign contracts Collect basic information Request for more data Explain final document Pre Approval process; set up workflow / account responsibility Line of interaction Onstage actions Line of visibility Backstage actions Line of internal interaction Run formal credit scoring model Support processes Prof. Christian Terwiesch Source: Yves Pigneur

Process Mapping / Service Blue Prints How to Redesign a Service Process Move work off the stage Example: online check-in at an airport Reduce customer actions / rely on support processes Example: checking in at a doctor’s office Instead of optimizing the capacity of a resource, try to eliminate the step altogether Example: Hertz Gold – Check-in offers no value; go directly to the car Avoid fragmentation of work due to specialization / narrow job responsibilities Example: Loan processing / hospital ward If customers are likely to leave the process because of long wait times, have the wait occur later in the process / re-sequence the activities Example: Starbucks – Pay early, then wait for the coffee Have the waiting occur outside of a line Example: Restaurants in a shopping malls using buzzers Example: Appointment Communicate the wait time with the customer (set expectations) Example: Disney Prof. Christian Terwiesch

Response Time Loss Models Prof. Christian Terwiesch

Different Models of Variability Waiting problems Utilization has to be less than 100% Impact of variability is on Flow Time Pure waiting problem, all customers are perfectly patient. Loss problems Demand can be bigger than capacity Impact of variability is on Flow Rate All customers enter the process, some leave due to their impatience Customers do not enter the process once buffer has reached a certain limit Same if customers are patient Customers are lost once all servers are busy Same if buffer size=0 Same if buffer size is extremely large Variability is always bad – you pay through lower flow rate and/or longer flow time Buffer or suffer: if you are willing to tolerate waiting, you don’t have to give up on flow rate Prof. Christian Terwiesch

Analyzing Loss Systems Resources 3 trauma bays (m=3) Ambulances / Helicopters Demand Process One trauma case comes in every 3 hours (a=3 hours) Trauma center moves to diversion status once all servers are busy incoming patients are directed to other locations Service Process Patient stays in trauma bay for an average of 2 hours (p=2 hours) a is the interarrival time p is the service time Exponential interarrival times Can have any distribution What is Pm, the probability that all m resources are utilized? Prof. Christian Terwiesch

Analyzing Loss Systems: Finding Pm(r) m • Define r = p / a • Example: r= 2 hours/ 3 hours r=0. 67 • Recall m=3 • Use Erlang Loss Table r=p/a • Find that P 3 (0. 67)=0. 0255 Given Pm(r) we can compute: • Time per day that system has to deny access • Flow units lost = 1/a * Pm (r) Prof. Christian Terwiesch

Implied utilization vs probability of having all servers utilized: Pooling Revisited Probability 0. 6 that all servers are utilized 0. 5 0. 4 m=1 m=2 0. 3 0. 2 m=3 0. 1 m=5 m=10 m=20 0 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 Implied utilization Prof. Christian Terwiesch 0. 8 0. 9 1 1. 1

Erlang Loss Table Probability{all m servers busy}= Prof. Christian Terwiesch

Response Time Review Prof. Christian Terwiesch

(My-law. com) My-law. com is a recent start-up trying to cater to customers in search of legal services online. Unlike traditional law firms, My-law. com allows for extensive interaction between lawyers and their customers via telephone and the Internet. This process is used in the upfront part of the customer interaction, largely consisting of answering some basic customer questions prior to entering a formal relationship. In order to allow customers to interact with the firm’s lawyers, customers are encouraged to send e-mails to my-lawyer@My-law. com. From there, the incoming e-mails are distributed to the lawyer who is currently “on call. ” Given the broad skills of the lawyers, each lawyer can respond to each incoming request. E-mails arrive from 8 A. M. to 6 P. M. at a rate of 10 e-mails per hour (coefficient of variation for the arrivals is 1). At each moment in time, there is exactly one lawyer “on call, ” that is, sitting at his or her desk waiting for incoming e-mails. It takes the lawyer, on average, 5 minutes to write the response e-mail. The standard deviation of this is 4 minutes. a. What is the average time a customer has to wait for the response to his/her e-mail, ignoring any transmission times? Note: This includes the time it takes the lawyer to start writing the e-mail and the actual writing time. b. How many e-mails will a lawyer have received at the end of a 10 -hour day? c. When not responding to e-mails, the lawyer on call is encouraged to actively pursue cases that potentially could lead to large settlements. How much time on a 10 -hour day can a My-law. com lawyer dedicate to this activity Prof. Christian Terwiesch

Jim’s Computer Jim wants to find someone to fix his computer. PC Fixers (PF) is a local service that offers such computer repairs. A new customer walks into PF every 10 minutes (with a standard deviation of 10 minutes). PF has a staff of 5 computer technicians. Service times average around 40 minutes (with a standard deviation of 40 minutes). JC 1. If Jim walks into PF, how long must he wait in line before he can see a technician? (Only include the waiting time, not any service time) JC 2. How many customers will, on average, be waiting for their computer to be fixed? Prof. Christian Terwiesch

Real Compute Real. Compute offers real-time computing services. The company owns 4 supercomputers that can be accessed through the internet. Their customers send jobs that arrive on average every 4 minutes (inter-arrival times are exponentially distributed and, thus, the standard deviation of the inter-arrival times is 4 minutes). Each job takes on average 10 minutes of one of the supercomputers (during this time, the computer cannot perform any other work). Customers pay $20 for the execution of each job. Given the time-sensitive nature of the calculations, if no supercomputer is available, the job is redirected to a supercomputer of a partner company called On. Comp, which charges $40 per job to Real Compute (On. Comp always has supercomputer capacity available). RC 1. What is the probability with which an incoming job can be executed by one of the supercomputers owned by Real. Compute? RC 2. How much does Real. Compute pay on average to On. Comp (in $s per hour)? Prof. Christian Terwiesch

Contractor A contractor building houses and doing renovation work has currently six projects planned for the season. Below are the items, and the estimated times to complete them: New construction at Springfield Bathroom remodeling at Herne Training time for solar roof installation Update web-site Renovation of deck at Haverford New kitchen at Rosemont - 60 days - 10 days - 2 days - 6 days - 8 days - 20 days Suppose the contractor starts immediately with the first project, no other projects get added to this list, and the contractor sequences them so as to minimize the average time the project waits before it gets started. What will the contractor be doing in 30 days from the start date of the first project? Prof. Christian Terwiesch

Call Center Consider a call center that has a constant staffing level. Because of increased demand in the morning, the call center has a very high utilization in the morning and a very low utilization in the afternoon. Which of the following will decrease the average waiting time in the call center? (a) Add more servers (b) Decrease the service time coefficient of variation (c) Decrease the average service time (d) Level the demand between the morning hours and the afternoon hours (e) All of the above Prof. Christian Terwiesch