1. 1 BIS 3106: Business Process Management (BPM) Lecture Nine: Quantitative Process Analysis (2) Makerere University School of Computing and Informatics Technology Department of Computer Science SEM I 2015/2016

2. 2 Business Process Analysis

3. 3 Process Analysis Techniques Quantitative Analysis Quantitative Flow Analysis Queuing Theory Process Simulation

4. 4 Why flow analysis is not enough? Flow analysis does not consider waiting times due to resource contention Queuing analysis and simulation address these limitations and have a broader applicability

5. 5 Queuing Theory Queuing theory is a collection of mathematical techniques to analyze systems that have resource contention. Resource contention leads to queues - we have all experienced in supermarket check-out counters, at a bank’s office, post office or government agency. Queuing theory gives us techniques to analyze important parameters of a queue, e.g., expected length of the queue expected waiting time of an individual case in a queue.

6. 6 Capacity problems are very common in industry and one of the main drivers of process redesign Need to balance the cost of increased capacity against the gains of increased productivity and service Queuing and waiting time analysis is particularly important in service systems Large costs of waiting and of lost sales due to waiting Prototype Example – ER at a Hospital Patients arrive by ambulance or by their own accord One doctor is always on duty More patients seeks help  longer waiting times  Question: Should another Medical Doctor position be instated? Why is Queuing Analysis Important?

7. 7 Delay is Caused by Job Interference If arrivals are regular or sufficiently spaced apart, no queuing delay occurs Queuing occurs every time user demand exceeds server capacity. Delays are due to several reasons. Let’s examine some of them… Deterministic traffic

8. 8 Burstiness Causes Interference Queuing results from variability in service times and/or interarrival intervals

9. 9 Job Size Variation Causes Interference Arrivals are deterministic (e.g. the intervals between 2 consecutive arrivals are always the same), however some jobs are longer than others Variable Job sizes

10. 10 High Utilization Exacerbates Interference The queuing probability increases as the load increases Utilization close to 100% is unsustainable  too long queuing times

11. 11 Queuing Theory A queuing system consists of one or multiple queues and a service that is provided by one or multiple servers. The elements inside a queue are called jobs or customers, depending on the specific context. E.g., in the case of a supermarket, the service is that of checking out. This service is provided by multiple cashiers (the servers).

12. 12 Queuing Theory While in the case of a bank office, the service is to perform a banking transaction, the servers are tellers. There is a single queue that leads to multiple servers (the tellers). Two examples illustrate an important distinction. Have multi-line i.e. multi-queue queuing systems (like the supermarket. Have single-line queuing systems (like the bank office).

13. 13 Example – Two Queue Configurations Servers Multiple Queues Servers Single Queue

14. 14 Queuing Theory Queuing theory provides a very broad set of techniques to analyze important parameters of a queue. Queuing theory models that are useful when analyzing business processes or activities within a process.

15. 15 A Commonly Seen Queuing Model (I) C C C … C Customers (C) C S = Server C S C S Customer =C The Queuing System The Queue The Service Facility

16. 16 Queuing Theory: Arrival Rate In the two models there is a single queue (single-line queuing system). Customers come at a given mean arrival rate: Same concept of arrival rate that we discussed when presenting Little’s law. For example, we can say that customers arrive at the bank office at a mean rate of 20 per hour.

17. 17 Queuing Theory: Arrival Rate For example, we can say that customers arrive at the bank office at a mean rate of 20 per hour. This implies that, on average, one customer arrives every 3 minutes ( hour). This latter number is called the mean inter-arrival time. If is the arrival rate per time unit, then is the mean inter-arrival time.

18. 18 Arrival Rates It would be strange to think that the time between the arrival of two customers at the bank office is always 5 minutes. This is just the mean value. Customers arrive independently from one another, so the time between the arrival of one customer and the arrival of the next customer is completely random. The arrival time of first customer and arrival of second customer can be random and has nothing to do with the arrival of third customer.

19. 19 Arrival Rates Such an arrival process is called a Poisson process. Distribution of arrivals between any two consecutive customers follows a so-called negative exponential distribution with a mean of In practice, the Poisson process and the exponential distribution describe a large class of arrival processes that can be found in business processes.

20. 20 Arrival Rates Poisson process will be used to capture the arrival of jobs or customers into a business process or an activity in a business process. The Poisson process can also be observed for example when we examine how often cars enter a given segment of a highway, or how often calls go through a telephone exchange.

21. 21 Processing Times Exponential distributions are useful when describing the processing time of an activity. An activity’s processing time is exponentially distributed. E.g., the amount of time it takes for a mechanic to make a repair on a car. Most repairs are fairly standard, and the mechanics might take for example one hour to do them. However, some repairs are very complex, and in such cases, it can take the mechanic several hours to complete.

22. 22 Processing Times Consider a doctor receiving patients in an emergency room. A large number of emergencies are quite standard and can be dispatched in less than an hour, but some emergencies are extremely complicated and can take hours to deal with. So it is likely that such activities will follow an exponential distribution.

23. 23 Service times as well as interarrival times are assumed independent and identically distributed If not otherwise specified Commonly used notation principle: A/B/C A = The interarrival time distribution B = The service time distribution C = The number of parallel servers Example: M/M/c Queuing system with exponentially distributed service and inter- arrival times and c servers A Commonly Seen Queuing Model (I)

24. 24 M/M/1 Queue In queuing theory field, a single-queue system is called an M/M/1 queue. Inter-arrival times of customers follow an exponential distribution (M) Processing times follow an exponential distribution (M) One single server (1) Jobs are served on a First-In-First-Out (FIFO) basis. In the case of M/M/1 queue, we also assume that when a job arrives, it enters the queue and it stays there until it is taken on by the server.

25. 25 M/M/1 Queue M/M/1 queue can be defined by means of the following parameters: A: Arrival Rate: is the average number of customers arriving per time unit. The mean inter-arrival time is then B: Service Rate: the mean number of customers that can be served per time unit. The mean processing time per job is then E.g. = 6 means six jobs are served per hour, that is, one job is served in 10 minutes (on average)

26. 26 M/M/1 Queue Given parameters and we can already state how busy a server is. This is called the occupation rate or resource utilization How busy a resource is at a given time. Occupation Rate is calculated as: If = 5 and = 6 then the occupation rate This value is relatively high.

27. 27 M/M/1 Queue A system with a high occupation rate of more than 100% is unstable. The queue will become longer and longer forever because the server cannot cope with all the demand.

28. 28 Queuing theory: basic concepts Basic characteristics:  (mean arrival rate) = average number of arrivals per time unit  (mean service rate) = average number of jobs that can be handled by one server per time unit time between arrivals and service time follow a negative exponential distribution c = number of servers (c =1) FIFO arrivals waiting service  c

29. 29 Structure of an M/M/1 system parameters

30. 30 Structure of an M/M/1 system parameters Given an M/M/1 system, queuing theory allows us to calculate the following parameters: 1. Lq —The average number of jobs (e.g. customers) in the queue. 1. Wq —The average time one job spends in the queue.

31. 31 Structure of an M/M/1 system parameters 3. W —The average time one job spends in the system. This includes both the time the customer spends in the queue and time the customer spends being serviced. 3. L —The average number of jobs in the system (i.e. the Work-in-Progress referenced in Little’s law).

32. 32 Electronic Design Company Example A company designs customized electronic hardware for a range of customers in the high-tech electronics industry. The company receives orders for designing a new circuit and on average it receives 0.05 orders per day The team of engineers design 0.1 orders of hardware per day.

33. 33 Electronic Design Company Example Problem can be mapped to an M/M/1 model assuming that the arrival of designs follows a Poisson process. Assume that the distribution of times for designing a circuit follows an exponential distribution. New design requests are handled on a FIFO manner

34. 34 Design Company The working day is taken as the time unit. On average, 0.05 orders are received per day On average 0.1 orders are fulfilled per day The occupation rate of this system

35. 35 Design Company Using the formulas for M/M/1 models, we can deduce that the average length of the queue: The average time an order spends on the queue is:

36. 36 Design Company It takes on average order 20 working days for an order to be fulfilled.

37. 37  Situation Patients arrive according to a Poisson process with intensity. The service time (the doctor’s examination and treatment time of a patient) follows an exponential distribution,   The ER can be modeled as an M/M/c system where c=the number of doctors Example – ER at County Hospital  Data gathering  = 2 patients per hour  = 3 patients per hour

38. 38 Interpretation To be in the queue = to be in the waiting room To be in the system = to be in the ER (waiting or under treatment) Is it warranted to hire a second doctor? Queuing Analysis – Hospital Scenario CharacteristicOne doctor (c=1)Two Doctors (c=2)  2/31/3 LqLq 4/3 patients1/12 patients L2 patients3/4 patients WqWq 2/3 h = 40 minutes1/24 h = 2.5 minutes W1 h3/8 h = 22.5 minutes

39. 39 With c = 1, we handle 3 patients per hour With c = 2, we handle 6 patients per hour. According to the results, the resource utilization is reduces when we hire two doctors and this reduces the length of the queue Queuing Analysis – Hospital Scenario

40. 40 Generally not applicable when system includes parallel activities. Requires case-by-case mathematical analysis. Assumes “steady-state” (valid only for “long-term” analysis) Process simulation is more versatile (also more popular) Drawbacks of Queuing Theory