DECISION MODELING WITH Prentice Hall Publishers and MICROSOFT EXCEL Chapter 15 QUEUING Part 2 Copyright 2001 Prentice Hall Publishers and Ardith E. Baker

MODEL 2: A FINITE QUEUE (WATS LINES) In this model, we attempt to select the appropriate number of _______lines for St. Luke’s. The telephone company can provide a great deal of _______in these matters, since queuing models have found extensive use in the field of _________ traffic engineering. This problem is typically attacked by using the _______model, “with blocked customers cleared.” This is a _____________queue with s servers, exponential interarrival times for the calls and a general distribution for the ______________(length of call).

“Blocked customers cleared” means that when an _______finds all of the servers __________(all of the lines busy), he or she does not get in a queue but simply________. Probability of j Busy Servers The problem of selecting the appropriate number of _____(servers) is attacked by computing the steady-state ______________that exactly j lines will be busy. This will be used to calculate the steady-state probability that all s _______are busy. Clearly, if you have s lines and they are all busy, the next _________will not be able to place a call.

The steady-state probability that there are exactly j busy ________ given that s lines (servers) are available is: Pj = (l/m)j /j! S k=0 s (l/m)k /k! where l = _______rate (the rate at which calls arrive) 1/m = mean _______time (the average length of a conversation) s = number of _________(lines) This expression is called the ____________Poisson distribution or the ________loss distribution. The value of Pj depends only on the _______of this distribution.

S Consider a system in which l = 1 (calls arrive at the rate of 1 per minute) 1/m = 10 (the average length of a conversation is 10 minutes) Here, l/m = 10. Suppose there are five lines in the system (s = 5) and we want to find the steady-state probability that exactly two are busy (j = 2). P2 = (l/m)2 /2! S k=0 5 (l/m)k /k! P2 = (10)2 /2•1 1 + 101/1 + 102/2•1 + 103/3•2•1 + 104/4•3•2•1 + 105/5•4•3•2•1 On the average, two lines would be busy 3.4% of the time. P2 = 50 1477.67 = 0.034

An ___________way of obtaining Pj that is easy to implement in a spreadsheet is as follows: Pi = Pi-1 (l/m)/i So, for example, once we know P2, we can ________ P3 as: P3 = P2(10)/3 = (0.034)(10)/3 = 0.1133 Likewise, P4 is found as: P4 = P3(10)/4 = (0.1133)(10)/4 = 0.2833 Each _________Pi-1 is multiplied by (l/m) and divided by i to achieve the new Pi.

The more interesting question is: “What is the ___________that all of the lines are busy?” since in this case, a potential caller would not be able to place a call on the _______lines. To find the answer to this question, we simply set ______(in our example s = 5) and we obtain P5 = P4(10)/5 = (0.2833)(10)/5 = 0.564 Or on the average the system is totally _________ 56.4% of the time.

Using the spreadsheet, the probability that a customer _______with 5 servers is easily calculated. We can then build a data table to determine this probability for different _________of s.

Enter the values for s, ranging from 0 to 10. Specify the formula (= F13) for the quantity that we want to track (the prob. that a customer balks). Highlight the cell range B22:C33 and click Data – Table. Specify $E$4 as the Column Input and click OK.

Column D calculates the __________improvement in this probability as servers are added. It is clear that the marginal effect of adding more servers__________.

Average Number of Busy Servers This quantity is called the__________ Average Number of Busy Servers This quantity is called the__________. If we define N as the average number of busy servers, then N = (l/m)(1 – Prob. that a customer will balk) Assume for this model, l = 1 and 1/m = 10. Thus, if 10 lines are_________, the probability that all 10 are busy is 0.215 (from previous table). It follows then that N = (10)(1 – 0.215) = 7.85 After finding N, the server ____________can be calculated by dividing N by s (the number of servers). Thus, 7.85/10 = 78.5%. Each server is busy 78.5% of the time and ______21.5% of the time.

MODEL 3: THE REPAIRPERSON MODEL Now we must decide how many repairpersons to hire to _________20 pieces of electronic equipment. Machines are _________on a first-come, first-served basis and a _______repairperson treats each broken machine. The failed machines form a ________in front of the multiple servers (repairpersons). This is an ______model, but it differs from the blood-testing model in that there is a _______number of items (20) that can join the queue.

Queuing models in which only a ________number of “people” are eligible to join the queue is said to have a finite_______________________. Models with an ______________number of possible participants are said to have an __________calling population. Consider the model with 20 machines and 2 repair-persons. Assume that when a machine is running, the time between ______________has an exponential distribution with parameter l = 0.25 per hour. Thus, the average ____between breakdowns is 1/l = 4 hours.

The time it takes to ______a machine has an exponential distribution and the _______repair time (1/m) is 0.50 hour. This model is an M/M/2 model with a ____________of 18 items in the ______and a finite calling population. In this case, the general equations for the steady-state probability that there are _______in the system is a function of l, m, s, and N (the number of _____________). Pn = N! n!(N – n)! (l/m)n P0 for 0 < n < s for s < n < N Pn = N! (N – n)!s!sn-s (l/m)n P0

S n=0 N Pn = 1 We also know that We thus have N + 1 ________equations in the N + 1 variables of interest (P0, P1, …, Pn). Although_____________, this makes it possible to calculate values of Pn for any particular model. There are, however, no simple ____________for the expected number of jobs (broken machines) in the system or for___________. If the values for Pn are computed, then it is a simple task to find a _____________value for the expected number in the system. You must just calculate: S n=0 N nPn expected number in system = L =

A spreadsheet can be used to compute values of Pn. NOTE: when you enter the value for the arrival rate (l) in the “MMs” worksheet, you need to enter N*l (the entire population’s arrival rate).

TRANSIENT vs STEADY-STATE RESULTS: ORDER PROMISING In this section, we will consider a situation in which we are interested in the _________(not steady-state) behavior of the system. _____________processes can be viewed as complex queuing systems. In fact, queuing systems __________is probably the most frequently used management science tool in manufacturing. SONOROLA company is concerned about when to ________a new customer order. The order is for 20 units of an item that requires __________processing at 2 work stations.

(10 units x 4 hrs/unit + 4 hrs + 4 hrs)  8 hrs/day = 10.5 days The average _______to process a unit at each work station is 4 hours. Each work station is ________for 8 hours every working day. By considering when the last of the 20 units will be ___________, it is estimated that it will take 10.5 days to process the order. The last unit must wait at Work Station 1 for the first 19 units to be completed, then it must be _________at Work Station 1, then at Work Station 2. Assuming that it does not have to wait when it gets to Work station 2, we can calculate the following: (10 units x 4 hrs/unit + 4 hrs + 4 hrs)  8 hrs/day = 10.5 days However, this analysis is somewhat___________. It ignores the ___________of the processing times and the possibility of queuing at Work Station 2.

The 4-hour _____________time at each work station was arrived at by __________many processing times that were less than 4 hours with a few processing times that were significantly ________than 4 hours (due to equipment failures at a work station while processing a unit). Next, check to see whether the _____________of the basic queuing model are met. The output from Work Station 1 are the _________to Work Station 2, and the time between arrivals is _____________because the processing time at Work Station 1 is exponential. The service time at Work Station 2 is ___________ because it is the same as the ______________time.

The units are processed on a_________, first-served basis at Work Station 2 and there is sufficient _____ capacity between the work stations so that the queue size is_________. However, the __________of an infinite time horizon is not met. We are only interested in the ________of the system until “________” 20 ends its processing. Let’s apply the basic model anyway and use it as an _________________. The time it takes to process 20 units is approximated as follows: 1. The last unit in the batch of 20 is estimated to leave Work Station 1 after 20 x 4 = 80 hours.

2. This unit will then wait in the _______in front of Work Station 2. 3. Finally, it will ________processing at Work Station 2, at which time all 20 units will have been completed. The total time that the last unit spends at Work Station 2 is W. Thus, our __________is 20 x 4 x W. Remember, for the basic model, W = 1/(m – l), for m > l. The problem is that m and l are ______(1 unit per 4 hours).

A spreadsheet can be used to simulate the ______of the 20 units through the 2 work stations. Assume that raw material is always ____________at Work Station 1 so that the next unit at Work station 1 can start as soon as the ___________unit is finished. This means that for Work Station 1, the start time of a unit is the __________of the previous unit. The start time of a unit at Work station 2 is either the stop time of ________on Work Station 1 or the stop time of the previous unit on Work Station 2, whichever is________. The stop time of a unit is just the ____________plus the _____________time. The finish time in days is calculated by dividing the stop time at Work Station 2 of the last unit by the number of___________.

The finish time is calculated to be 10 The finish time is calculated to be 10.5 days if every unit takes exactly 4 hours at every work station.

To analyze the impact of __________time variability, replace the _________processing time of 4 at Work Station 1 in the spreadsheet with the appropriate ________distribution (exponential with a mean of 4). In @RISK, enter =RiskExpon($B$1) to make the _______time a random variable. We would like to know the 99th ___________of this random variable so that we could then promise the order in that number of days and be _____sure that it would actually be _____________on time.

The following @RISK output shows the 99th percentile for cell F2 (the finish time in days) based on 1000 sets of 40 random processing times.

To be 99% sure of having the order completed by the _________date, set the due date to be 18.28 days after the material becomes ____________at Work Station 1. The ____________that takes place at Work Station 2 has increased the _________(time from the start of the order to its completion) by nearly 8 days (18.28 – 10.5) over what it would be if there were no __________in the processing times.

Here is a histogram of the finish time Here is a histogram of the finish time. Note that the time can vary from 6.5 days up to 21 days.

THE ROLE OF THE EXPONENTIAL DISTRIBUTION The role of the ___________distribution in ________ queuing models is useful in understanding the use of queuing models. Most analytic results for queuing situations involve the exponential distribution either as the distribution of ___________times or service times or both. The following three properties help to identify the set of _______________in which it is reasonable to assume that an exponential distribution will______.

1. _______________: In an arrival process, this property implies that the ____________that an ________will occur in the next few minutes is not influenced by when the last arrival occurred. This situation arises when (a) there are many ___________who could potentially arrive at the system (b) each person decides to arrive _____________of the other individuals (c) each individual selects his or her time of arrival completely at_________

2. ___________________: With an exponential distribution, small values of the ________time are common (as shown below). Prob S<t 1.0 0.632 10 20 30 40 t This graph shows the _____________that the service time S is less than or equal to t if the ________service time is 10.

The graph showed that more than 63% of the service times were _________than the average service time (10). Compare this to the ___________distribution where only 50% of the service times are ___________than the average. The practical implication is that an exponential distribution can best be used to model the distribution of _______________in a system in which a large proportion of “jobs” take a very ___________and only a few “jobs” run for a long time.

3. Relation to the __________________: While introducing the_____________, a relationship between the exponential and Poisson distributions was noted. In particular, if the time between arrivals has an ___________distribution with parameter l, then in a specified period of time (say, T) the number of arrivals will have a ___________ distribution with parameter lT. Then, if X is the number of arrivals during the time T, the probability that X equals a specific number (say, n) is given by the equation Prob [X = n] = e-lT(lT)n n! For any _____________integer value of n.

The _____________between the exponential and the Poisson distributions plays an important role in the theoretical ______________of queuing theory. It also has an important practical________________. By comparing the number of _______that arrive for service during a specific period of time with the number that the Poisson distribution_____________, the manager is able to see if his or her choices of a model and ____________values for the arrival process are reasonable.

QUEUE DISCIPLINE In addition to the _________distribution, service distribution and number of servers, the queue __________must also be specified to define a queuing system. So far, we have always assumed that arrivals were served on a first-come, first-serve basis (often called __________, for “first-in, first-out”). However, this may not always be the case. For example, in an elevator, the last person in is often the first out (___________). Adding the possibility of selecting a ________queue discipline makes the queuing models more ____________. These models are referred to as __________models.