PHILIPPINE WOMEN’S UNIVERSITY

Waiting Line Theory
A Narrative Report

2014

MBA PROGRAM

Introduction Each organization has different illustrations of waiting lines or commonly called “queue”. No wonder why we also need to study waiting line because it is part of our everyday routine as well. It merely affects the performance and profit of the company. Waiting line or queue refers to a busy service facility or server so service is momentarily occupied or being used. One great example is during enrollment. At our very first step at the school gate, you will fall in line to have your bag checked by the security personnel. You will also need to fall in line at your college department to seek advise of what subjects needed to take. You might wait for your adviser attending to other students. At the time you have your subjects needed to enrol, you will need to fall in line at Registrar’s office for subjects encoding. Once you get your assessment form, you will be redirected to Accounting Office for payment and expect a new line again. Then, you will need to update your school ID card after payment, it means another line. Waiting line or queue is a repetitive scenario in our everyday lives. We can’t deny but we are also used in waiting. The organization itself also encounter waiting line or queue. One example is when an airplane has to wait in line for fueling, inspection, a particular gate, a specific flight route, an assigned crew, food loading, verified passenger count, taxi takeoff and landing clearance for each trip. All organizations are concerned with waiting lines as well because they value their time and hate to be idle since it means loss to the company. When waiting lines are poorly managed, this will result to customer dissatisfaction and worst, brand-switching.

According to the the book of Levin, Walt Disney is very familiar with waiting lines. Walt Disney introduced the “Folded Waiting Lines”. These folded lines help to make the long waiting lines look shorter than they actually are. Plus, the very strong rail that divides the lines make really diffcicult to leave the line once you are on the line. A.K. Erlang is a Danish Engineer who is associated with telephone industry for his pioneering work of queueing theory. Erlang was doing testing involving fluctuating demand for telephone line facilities and its effect upon automatic dialing equipments. Waiting-line models are helpful in mutually manufacturing and service parts. Exploration of queues in terms of waiting-line span, average waiting time, and other dynamics aids us to realize service systems such as bank teller stations, maintenance activities and shop-floor control activities. Indeed, patients waiting in a doctor’s office. Both utilize individual and tools resources to reinstate important manufacture resources (people and machines) to fine state. Simple Queuing System Figure 1 shows the Simple Queuing System that represents the operation of the checkout counter however in a Car Wash business type as shown in the figure. In a basic situation like in a convenience store, if the manager would like to reduce the length of the waiting line that would usually outline at the single checkout counter, another cash register must be opened and another checker can be added. If the queues are still too extensive, further counters can be added. Each accumulation, of course, increases its cost, but at the same time, each additional diminishes the instance customers have to wait for service. The manager tries to hit a happy medium where waiting lines are small sufficient to lessen client ill will, but it is obviously not

practical to afford such wide-ranging service amenities that no waiting line, or queue, can ever expand. With that, the manager aims to balance the increased cost of additional facilities to shorten the waiting lines against customer ill will which adds up as the average length of the queue increases.

There are so many applications of waiting lines. All of us experience waiting lines by ourselves so no wonder that we all know that we wanted not to wait too long. In our own homes, we encounter waiting line or queue. For example, if we only have one comfort room and the schedule of the family members are all the same, that means we have to wait until the person ahead of us goes out the comfort room because we can use it. We also experience waiting line or queue in our community. When riding into a jeepney or any public vehicle, we tend to wait until it reaches the loading station where we are waiting and we tend to fall in line in getting into that vehicle. Even a little kid experiences waiting line or queue, the little kid tends to wait for his or her school bus that fetch other students home before their batch. There are so

many industrial applications of waiting line or queue. The manager should consider the lost of time persons waiting for the service and the wages of each person providing the service. When an employee is idle, it is automatically a loss to the company because the company is paying each minute that the employee is on duty so it is expected for him or her to become productive and busy attending to his or her job however if the employees are idle, the company still pays for them. It means, the employees are getting paid even when they idle or not doing any task that could contribute profit to the company. Employees are at work are expected to remain busy performing their duties and responsibilities however it is not the real situation inside the workplace. There is always an idle time. Tool Crib System: Problem-Solving Levin indicates a Problem- Solving activity in his book: In the typical machine shop, the expensive cutting tools required in the machining processes are kept in a central location often referred to as a tool crib. This crib is staffed by one or more persons who check out the tools required by the machinists in the shop. The machinist who requires a certain tool proceeds to the tool crib, presents a tool authorization to the attendant and is issued the required tool. The machinist then goes back to work, does the job and finally goes back to the tool crib and checks in the tool; if required, the machinist then draws another for the next job to be performed. Because some of the tools involved in work of this kind are very expensive, this procedure is needed to ensure adequate control of the tool inventory. The machinists are idle when they wait in line at the tool crib service which is not their fault because they are really mandated to fall in line however it is also a loss to the company

because they are being paid. This type of loss is computable since it is basically the amount of per time workers are compulsory to wait multiplied by the wages they receive per hour. When the employees who staff the tool crib are idle when no machinist is requiring service, their wages is also a lost to the company. Kindly look at the table below to see the representation of this problem as example of Levin.

Figure 2 Tool Crib System under several alternatives The table is divided according to the number of attendants which are 1, 2, 3 and 4. The first row indicates the number of arrivals of machinists during 8-hr shift which is 100 for all divisions. On the other hand, the average time each machinist spends waiting for service (minutes) are different because it varies depending on the number of attendants. Of course, more attendants, less waiting time of the machinists and less idle because the queue is cut off

into certain number of attendants. Having one attendant, the machinists need to spend ten minutes before getting into the tool crib. Having two attendants, machinists need to spend six minutes, four minutes for having three attendants while 2 minutes when there are four attendants. The machinists lost 1000 minutes during their 8-hr shift under one attendant for the tool crib. Machinists lost 600 minutes for having two attendants. Moreover, they lost 400 minutes for having three attendants and lost 200 minutes when four attendants are present. Generally, the average hourly pay rate of the machinists is $12. If we are going to compute for the value in dollars of how much the machinists lost time is, the formula is $12 divided by 60 minutes then multiplied to lost time. With one attendant, that will be $12 / 60 minutes = 0.2 then, 0.2 * 1000 minutes = $200 so $200 is the value of the company loss for idle time. With two attendants, $12 / 60 minutes = 0.2 then, 0.2 * 600 minutes = $120 value of lost. With three attendants, $12 / 60 minutes = 0.2 then, 0.2 * 400 minutes = $80 company loss. With four attendants, $12 / 60 minutes = 0.2 then, 0.2 * 200 = $40 lost time. The hourly pay rate of the tool crib attendants is $6. The total pay of tool crib attendants for 8-hr shift is $48 for one attendant, $96 for two attendants because $48 multiplied to two then, thrice of $48 is 144 which is for three attendants and four times of $48 is $192 for having four attendants. We can compute for the machinists’ lost time plus tool crib attendants pay or the total costs by summing up the machinists lost time and tool crib attendant pay. $248 for one attendant because $200 which is the value of machinists lost time plus $48 of tool crib attendant pay. Next one is $216 which is derived from $120 + 96 then, $80 + $144 = $224 is the total cost when we have three attendants and $232 with four attendants which is derives from $40 + $192. Finally, we can

come up with the optimal number of tool crib attendants by selecting the lowest value of total cost which is having two tool crib attendants because it is only $216.

Figure 3 Tool Crib System under several alternatives with solution

In order to lessen the waiting time of the machinists is to supply ample tool crib attendants to avoid forming queue. Few attendants are needed sin ce the machinists arrive in group from time to time. It is considered a lost to the company when the tool crib attendants are idle if there is no machinist arrives for service unless they can do other tasks which are workrelated and within their scope. We should identify the most effective and appropriate solution which covers all the factors in the problem and figure out the ratio of tool crib attendants to service machinists in the lowest total cost. Of course, we should pick the lowest total cost

because it is our main target why we wanted to lessen the queue. The longer the queue, the higher cost the company will incur. If we are going to examine the figure 2 above, we can understand the illustration and provided options under observation over long period of time which is the 8-hr shift by putting several different staffing alternatives. Figure 3 merely represents that having two attendants is the optimal or best solution since it has the lowest total cost that the company might incur. It will both minimize the machinists lost time and the crib attendants wages. Hence, having more one or more than two attendants will increase the total cost. In the real set up of business industry, every company wants the lowest total cost and making each employee to become productive as they can be. Wasting of time, energy, money and workforce is very redundant and pointless so we should be able to understand and learn different analytic techniques regarding waiting line. The given example of the tool crib system is only one of many applications of waiting line that a manager should consider. The manager should always come up with the optimal solution and take this as an opportunity to increase the profit of the company. Queuing Objectives and Cost Behavior Once again, the main objective of the example earlier was to minimize the total cost by cutting off the lost time by the arrivals waiting for service and providing the service. As what we can observed from the tool crib problem, the lost time of machinists decreased having more tool crib attendants however it affects the total cost that the company will incur. In other words, the lost time is a great factor for the overall total cost. Each factor or variable affects the totality of the business operation. Moving forward, Figure 4 gives us more examples of queues.

Situation Airport Restaurant Class Registration Hospital Post Office Telephone Exchange Gasoline Station Job Interview Court System Intersection

Arrivals Airplanes Customers Students Patients Letters Incoming Calls Motorists Applicants Cases Motorists

Queue Planes stacked up Customers waiting to eat

Service Facility Runway Tables/Waiters

Students waiting to register Registrars Patients waiting to use facility Letters waiting for distribution Persons making calls Motorists waiting for gasoline Applicants waiting to see interviewer Untried cases Cars backed up Rooms/Physicians Sorting System Switching System Pumps/Personnel Interviewer Judge/Courtroom Traffic Light/ Highway Capacity

Figure 4 Examples of Queues We can also see the basic relationships between and among the elements involved in queuing problems which can be illustrated graphically. In Figure 5, it is a graphic representation showing the relationship between level of service and cost of waiting time. It explains that the higher the level of service or having more counters or service facilities are provided, the cost of time spent waiting for the service is decreased. Of course, more service facilities can lessen the queue and that is one main objective of queuing theory however we must also consider the cost of putting more service facilities. Putting more service facilities may also increase the level of idle time by the attendants or employees waiting for the arrival of people needing their service. It is like a domino effect that one variable has something to do with the totality of the operations.

We should come up with good balance of these variables to lessen the loss that the company will incur.

Figure 5 Relationship between level of service and cost of waiting time The next figure would be the graphical representation of the relationship between the level of service and the cost of providing that service. It clearly tells us that the cost of providing a certain service will increase if the level of provided service increases. For example, more service facilities are provided will also boost the total cost that the company will incur. The manager should be keener in adding more service facilities just to lessen the queue. Yes, the queue will be lessened but it is pointless if the total cost is beyond expectations. It is not recommended to put more service facilities in operations with having a small to medium queue because this will result to more idle time. More idle time means more loss to the company. It is

derived from the cost of wages of the employees spent by the company during idle time and the cost of providing more service facilities.

Figure 6 Relationship between level of service and cost of providing service Figure 7 represents the combinations of the last two figures. It focuses on determining the optimal service level. Operations managers must distinguish the transaction that takes place between two costs: the cost of providing good service and the cost of customer or machine waiting time. Managers want queues that are short enough so that clients or customers do not become irate or dissatisfied and whichever leaves without any purchase but never come back. However, managers may be enthusiastic to permit various waiting if it is reasonable by a considerable savings in service costs. One means of assessing a service facility is to understand the total expected cost. Total cost is the sum of expected service costs plus expected waiting

costs. Managers in several service centres can be different in terms of proficiency by having reserve staff and equipment that they can allocate to particular service situations to avoid or cut down extremely long lines. Waiting cost may be a sign of lost productivity of workers while trappings or equipments lay ahead repairs or may plainly be an approximation of the cost of clients lost because of poor service and long queues. In some service systems, the cost of long waiting lines may be unbearably high.

Figure 7 Determining the Optimal Service Level

Standard Language and Definitions for Waiting Lines We will also study the different language and definitions related with waiting lines for us to totally understand the use and importance of waiting line models. We should start with the basic parts of queuing system which are calling population which is also called the arriving population or serviced population, next is queue or the waiting line itself and lastly, the service facility. Let’s tackle the first part which is the calling population. It has three characteristics which are the size, arrival characteristics and behaviour. Population sizes are measured either unlimited which is essentially infinite or limited or finite. When the quantity of clients or arrivals at a time at any specified instant is just a diminutive segment of all probable arrivals, the arrival population is considered unlimited or infinite. Examples of unlimited populations consist of cars arriving at a carwash, shoppers arriving at a supermarket, and students arriving to register for classes at university. An illustration of a limited or finite, population is set up in an arcade that has, say, eight gaming machines. Each of the population is a possible customer that may split behind and wants to play. We should remember that population is unlimited or infinite when a queue in which a practically boundless quantity of individuals or stuff could demand the services or in which the figure of clients or arrivals on hand at any certain flash is an incredibly diminutive fraction of prospective arrivals. On the other hand, Limited or finite population is when a queue in which there are only a limited number of latent customers of the service facility. Next characteristic is the arrival characteristics or the pattern of arrivals in the system. Clients appear at a service facility either based to some identified allotted time which is for

example, one customer every 15 minutes or one student every half hour or else they arrive randomly or in no particular pattern. Arrivals are considered casual or unsystematic when they are free and independent of one another and their presence cannot be forecasted accurately. Repeatedly in queuing problems, the amount of arrivals per component of time can be approximated by a chance distribution known as the Poisson distribution which will be discuss later on. Last characteristic is the behaviour of the calling population. Most queuing models believe that an arriving patron is a patient consumer. Patient customers are people or machines that stay in the queue until they are provided and served without switching between lines. Regrettably, life is complex by the reality that individuals have been identified to balk or to renege. Customers who balk are those who decline to adhere along the waiting line because it is too lengthy to go well with their desires or comfort. Reneging customers are those who go through the line but then turn out to be irritated or impatient and depart without finishing their transaction. In fact, both of these circumstances just give out to emphasize the call for queuing theory and waiting line study. Moreover, jockeying is when someone moves back and forth and jumped among multiple queues. The characteristic of the queue or the waiting line itself is the second module of a queuing system. The extent of a line can be classified as limited or unlimited. A queue is limited when it cannot, either by law or for the reason that of physical boundaries rises to a countless span. A small computer shop, for example, will have only a limited number of working computers so it could form a queue when the arrival of customers is larger than the number of working computers. A queue is unlimited when its size is unobstructed, as in the case of the toll booth

serving arriving drivers in an expressway. Most of us tend to think that the other line always moves faster than where are we currently in. We also assume that if we change lines, the one we just left will start to move faster than the current line we are in. Last part is the service facility and we are going to discuss its characteristics. The physical layout of the queuing system is described by the number of channels which is also called the number of servers. Service systems are usually classified in terms of their number of channels or number of servers and number of phases which is the number of service stops that must be completed to complete the transaction and exiting the system. A single-channel queuing system, with one server, is typified by a convenience store with only one cashier. If the convenience store has several cashiers on duty, with each customer waiting in one common line for the first available cashier, then we would have a multiple-channel queuing system. Most convenience stores today are multichannel service systems, as are most large hair salon, ticket booths and remittance centers. In a single-phase system, the customer receives service from only one service facility and then exits the system. One example is when a liquor shop in which the person who takes your order and also brings your item and takes your money is a single-phase system. So is when HR personnel or recruitment specialist in which the staff taking your application will also grades your test and will hire or reject you. However, say the registrar’s office in any University requires you to place your registration form at Encoding station, pay at Accounting Office and pick up your validated registration form at a third. In this case, it is a multiphase system. Likewise, if the registrar’s office is large or busy, you will probably have to wait in one line to complete your subjects’ approval and encoding of subjects on the first service stop, queue again to pay your

tuition fee and finally go to a third counter to validate your registration form and have your school ID stamped.

Another characteristic of Service Facility is the Queue Discipline in which unit in the calling population receives service. The classifications are Priority and First Come, First Served. Priority Discipline has two subclassifications which are preemptive and nonpreemptive. Preemptive discipline allows the members of the calling population to interrupt the members who are already receiving the service. For example, if your company director who will also take his or her lunch the same time as yours and will join the same queue as yours for buying the lunch meal and will ask you to let him or her be the first on the line because he or she is in a hurry because there’s a scheduled important meeting with the board members in few moments. Nonpreemptive discipline of queue is considered that those who have highest authority is automatically will be the first in line of being served. First come, first served has neither authorities nor priorities which is applicable in most public situations. It is simply serving the first and foremost in the line and followed by the sequential order in the line. Combinations of these

queue disciplines are also being practiced nowadays. For example in supermarket, shoppers with less than 10 items have their own line however they also practice first come, first served. Shoppers who are senior citizens or disabled have their own queue as well. Shoppers who have many carts have their own queue. Shoppers who have loyalty card have their own line. Shoppers who are using Debit Card or Credit Card for payments have their own line as well. The next characteristic is the appropriate probability distribution describing service times. In line with this, we should identify the best probability to describe the behaviour the service times. Service patterns are like arrival outlines in that they may be either stable or random. If service time is steady, it takes the equal sum of time to take care or assist each customer. This is the case in a machine-performed service process such as an automatic loading machine. More frequently, service times are randomly distributed. Elementary queuing system: constant arrival and service times Example 1: No Queue, Idle Time It only means that there are sufficient service facilities to serve the arrival of calling population. It’s like when there are only average of three customers arrive at the bank per hour however there are also five available bank tellers present per hour so there will be no queue because all customers are being served at the same time but there’s an idle time because available bank tellers who are waiting for other customers are not busy. Example 2: No Queue, No Idle Time For example the constant rate of customers is 10 per hour and the time interval is 6 minute during that hour. Let’s also consider that a service can be performed at a constant rate of 10 per every hour. We can come up with no queue because the arrival rates are the same with

the same rate as the serviced level. There will also be no idle time because all the service facilities match the rate of the arrival of the calling population. It may be after completing one transaction; another customer will arrive exactly right after the transaction is completed. Example 3: Queue Forms, No Idle Time Let’s say that the arrivals occur at a constant rate of 10 per hour, occurring every 6 minutes during that hour. Assume also that services are performed at a constant rate of 8 per hour. So a queue will form because the arrival of calling population is higher than the number of service facilities and it has something to do with the number of minutes needed to complete one transaction. As customers arrive and joins the queue, the length of the queue increases as well. SINGLE CHANNEL QUEUING MODEL: POISSON DISTRIBUTED ARRIVALS AND EXPONENTIALLY DISTRIBUTED SERVICE TIMES There are several queuing models which will help us analyze the queuing systems. First is the Single Channel Queuing Model: Poisson-Distributed Arrivals and Exponentially

Distributed Service Times. When we say Poisson Distribution, it’s the behavior of arrivals as occurring at random. Exponential Distribution on the other hand is the distribution of the service times in a service facilty. It yields the distribution of the time intervals between arrivals. When we use these queuing models, there are conditions to be considered. For the Single Channel Queuing Model: Poisson Distributed Arrivals and Exponentially Distributed Service Times, here are the 7 conditions or assumptions to consider. 1. Arrival rates must be Poisson-distributed. 2. Service times are exponentially distributed.

3. The queue discipline is First Come, first served. 4. There is one channel. 5. The calling population is infinite. 6. Mean arrival rate is less than the mean service rate. This must be the case

otherwise, if the arrival rate is greater than the service rate, the queue will grow infinitely long and the service facility will fall behind. 7. Waiting space for customers in the queue is infinite. There are 5 equations used in this model.

Let us take the sample problem in the book. Metrolease Furniture Rental

Given: 1 loading dock with a 3-person crew λ = 4 trucks per hour μ= 6 trucks per hour (6 trucks x 1 crew) if there are 2 or 3 crews, the mean service rate is: µ = 12 trucks per hour (6 trucks x 2 crews) µ = 18 trucks per hour (6 trucks x 3 crews) $20 – truck cost per hour $ 6 – labor cost per hour Using the 5 equations above, just substitute the values and we have the ff: answers. The objective here is to determine if the company should add 2 or 3 crews and how much it will cost the company. Using the 5 equations above, just substitute the values and the answers are:

1 crew µ=6 Mean # of trucks in the queue Mean # of trucks in the system L L 1.333 2.000

2 crews µ = 12 .167 .500

3 crews µ = 18 .063 .286

q

s

Mean time in the queue by a truck Mean time in the system by a truck Prob. service facility is busy

W W P

q

.333 .500 .667

.042 .125 .333

.016 .071 .222

s

w

As you can see here, the average number of trucks in the queue (Lq) decreases as the number of crew increases. And the average time in the queue (Wq) decreases as the number of crew increases. We want to reduce the waiting time of our customers so adding more crew is a certainty. But we also have to consider the cost here if we add more crews. Now let us compute for the additional costs. To compute for the truck cost per day, we multiply the average number of trucks in the system by the total number of working hours per day and multiply it also by the truck cost. To compute for the crew cost per day, we multiply the number of members in a crew by the labor cost times the total number of working hours per day. Then, we add the truck cost per day and the crew cost per day to get the total cost per day.

TRUCK COST/DAY

CREW COST/DAY

TOTAL COST/DAY

1 crew

2 x 8hr x $20 = $320

3 x $6 x 8hr = $144

$ 464

2 crews

.5 x 8hr x $20 = $ 80

6 x $6 x 8hr = $288

$ 368

3 crews

.286 x 8hr x $20 = $ 46

9 x $6 x 8hr = $432

$ 478

For having 1 crew, truck cost per day is $320. We got this by multiplying 2 which is the average number of trucks in the system multiply it by 8 hours then multiply it also by the truck cost. For the crew cost, in 1 crew there are 3 members. So 3 times $6, which is the labor cost multiplied by 8 hours, a total of $144. For the 2 crews, total cost per day is $368 and for the 3 crews, total cost per day is $478. Looking at the total costs per day, adding 2 crews will reduce the total cost per day from $464 to $368. And from the first table above, the waiting time is also reduced, from .333 hours to .042 hours or 2.52 minutes (.042hrs x 60min). So the best decision here is to have 2 crews since not only is the waiting time reduced, but also the total cost per day is also reduced. SINGLE CHANNEL QUEUING MODEL: POISSON-DISTRIBUTED ARRIVALS AND ANY SERVICE TIME DISTRIBUTION Another queuing model is the Poisson-distributed arrivals and any service time distribution. As we have discussed before, Poisson distribution yields the number of arrivals in a specified period of time and considers the behavior of arrivals as occurring at random. The service time in this case is undefined and does not fit an exponential distribution. The conditions for this model are: 1. The service times are independent of each other. This means that the length of service for any other customer does not affect the service for any other customer. 2. The same distribution of service times applies to all customers.

3. The mean service time (1/µ) and the variance of the service time (σ 2) are known.

The other conditions are the same as those from the previous model.

There are 5 equations used in this model.

Note: When we say “system”, they are the customers waiting in the queue and also those customers being served. Let’s take the sample problem in the book. CNNB BANK PENNSYLVANIA STATE UNIVERSITY BRANCH Operating Hours: Mon – Fri 12nn – 4pm

Mon – Thurs – λ = 8 students/hr (poisson distributed) ; µ = 24 students/hr (expo. Dist) Fridays - λ = 34 students/hr (poisson distributed) ; µ = 40 students/hr (expo. Dist)

Teller’s weekly salary: $180 Complaints: The limited operating hours and the waiting time during Fridays. Objective: To find out whether ordering a 24 hour banking machine will reduce the waiting time. Banking Machine:$250 Mean service time = 90 seconds(per transactions) Standard deviation σ = 8 seconds  When we say arrival rate (λ), it is the expected number of customers who arrives in a given period. Service rate (µ) is the number of customers served in a specified time period. It is different form the service time. Service time is the amount of time it takes to serve each customer. It must be converted and expressed in service rates so that it will be compatible with the arrival rate. STEPS: 1. Solve for µ 1 hr = 60min 60sec 1 hr = 3600 sec 1min

90 sec µ = 40 students/hr

2. Solve for σ2 (variance) ; σ = 8 sec σ = 8 sec 1 min 60sec = 8/3600 = .002222 hrs σ2 = (.002222)2 = .000005 So we now have the mean service rate (µ) and the variance (σ2). And our mean service rate is compatible with our arrival rate, which is expressed in hours. And our variance is also expressed in a fraction of an hour. To attain our objective, which is to determine whether the use of banking machine will reduce the waiting time of the customers, we must compute first for the teller’s performance characteristics by using the first model, the exponentially distributed service times. This is because in the problem, the arrival rate is Poisson distributed and the service times are exponentially distributed. And all the other conditions are met also. We need to compute for the teller’s performance so that we can also compare it to the banking machine’s performance. 1 hr 60min

TELLER Mean # of students in the queue Mean # of students in the system Mean time in the queue by a student Mean time in the system by a student Prob. a service facility is busy Lq Ls Wq Ws Pw 4.817 5.667 .142 .167 .850

1. Lq = λ __ µ(µ-λ) =34 /(40-40-34) =4.817 2. Ls = λ____ µ-λ = 34/ (40-34) = 5.667 3. Wq = λ__ µ(µ-λ) = 34/ 40(40-34) = .142
2

2

4. Ws= 1__ µ-λ = 1 / (40-34) = .167 5. P w = λ / µ = 34 / 40 = .85

Now that we have the teller’s performance, the next step is to compute for the banking machine’s performance using the present model. Using the equations for the any service time distribution model, we substitute the values and we came up with these answers. 1. Lq = λ2σ2 + (λ/µ)2 2(1 - λ/µ) = (342)(.000005) + (34/40)2 4. Ws = Wq + 1/µ = .071 + 1/40 =.071 + .025

2(1- 34/40) = 2.428 2. Ls = Lq + λ/µ = 2.428 + 34/40 = 3.278 3. Wq = Lq / λ = 2.428 / 34 = .071

= .096

5. Pw = λ/µ = 34/40 = .850

TELLER Mean # of students in the queue Mean # of students in the system Mean time in the queue by a student Mean time in the system by a student Prob. a service facility is busy Lq Ls Wq Ws Pw 4.817 5.667 .142 .167 .850

BANKING MACHINE 2.428 3.278 .071 .096 .850

Let’s look at this table.

If you can see at the table above, there is a huge difference in the average waiting time in the queue (Wq), from .142 hours or 8.52 min to .071 hours or 4.26 min. The average waiting time is almost reduced by half. Although using a banking machine will increase our service cost, from $180 to $250, the objective here is to reduce the waiting time of the customers. So, the bank manager here will likely order a banking machine. Although it will cost the bank more to use a banking machine, waiting time and cost will be reduced. Waiting cost does not only refer to amounts and figures. It also refers to the number of customers lost, and the loss of a customer’s goodwill. Sometimes, losing a customer is even more expensive than buying an extra machine. Also, there are times when the solution to a queuing problem may require management to make a trade-off between the increased cost of providing better service and the decreased waiting costs derived from providing that service. SINGLE CHANNEL QUEUING MODEL: POISSON-DISTRIBUTED ARRIVALS, EXPONENTIALLY

DISTRIBUTED SERVICE TIMES AND FINITE WAITING CAPACITY A finite queue is when the length of the queue is limited because space may permit only a limited number of customers to enter the queue. Since there is a limit in the number of customers, this limit is represented by M. It is also the maximum number of customers in the system. All the conditions in this model are the same with the Single Channel: Poisson distributed arrivals and Exponentially distributed service times, except that in this case, the waiting capacity is finite. Unlike the other two models wherein they have an infinite waiting capacity.

There are 7 equations used in this model. 1. P0 – probability that no customers are in the system P0 = 1- (λ/µ)___ 1- (λ/µ)M+1 2. Pw – probability the service facility is busy Pw = 1-P0 3. PM – proportion of customers lost coz system is “FULL” PM = λ µ
M

P0

4. Ls = mean number in the system Ls = Pw – M(λ/µ)PM 1 – (λ/µ) 5. Lq = mean number in the queue

6. Ws = mean time in the system Ws = Ls λ(1-PM) 7. Wq = mean time in the queue

We have here a sample problem from the book. CNNB University Park Branch (Drive-in Branch) M = 3 in the system λ = 14 cars/hr Poisson dist. µ = 20 cars/hr Expo. Dist. The objective here is to find out the effects on the system performance if the number of spaces for cars is expanded to M=4 or M=5. Using the equations above, assuming that M = 3, we substitute the values and we came up with these answers. If M=3

1. P0 = 1- (λ/µ) 1- (λ/µ)M+1 2. Pw = 1-P0 = 1-.395

= 1-(14/20) 1-(14/20)3+1

= .3 .7599

= . 395 prob. system is idle

= .605 prob. the service facility is busy 3. PM = (λ/µ)M P0 = (14/20)3 (.395) = .135 proportion of customers lost because system is full 4. Ls = Pw – M(λ/µ)PM 1 – (λ/µ) = .605 -3(14/20)(.135) 1-(14/20) = .3215/.3 = 1.071 mean number of cars in the system 5.Lq = Ls – λ(1-PM) µ = 1.071 – 14(1-.135)

20 = 1.071 - .6055 = .465 mean number of cars in the queue 6. Ws =___Ls___ λ(1-PM) = 1.071____ 14(1-.135) = 1.071/12.11 = .088 mean time in the system by a car 7.Wq = Ws – 1 µ = .088 – (1/20) = .088 - .05 = .038 mean time in the queue by a car

We’ll still use the same equations if M=4 and M=5. Let’s look at the table below.

M=3

M=4

M=5

Prob the service is idle

P0

.395

.361

.340

Prob the service facility is busy

Pw

.605

.639

.660

Prop of customers lost coz system is full

PM

.135

.087

.057

Mean # of cars in the system

Ls

1.071

1.318

1.532

Mean # of cars in the queue

Lq

.465

.681

.872

Mean time in the system by a car

Ws

.088

.103

.116

Mean time in the queue by a car

Wq

.038

.053

.066

Looking at the table above, as M increases, the proportion of customers lost because system is full decreases. This is actually good since this is what we want to happen, to minimize the lost of customers who drive away because the system is full. But as for the others, as M gets larger, everything else increases. The average number of cars in the system increases, the average number of cars in the queue increases and so on. All except for the P M. This is always the case since making M larger increases the arrival rate, the number of customers. This is what we want to happen, to gain more customers and not to lose them. This is why we want to

expand in the first place, to have more customers to serve and avoid losing customers just because the space is limited. If we just base our decision from the proportion of customers lost because system is full or the PM, expanding to M=5 is the best solution here. But we also have to consider the expansion costs and it is not included in this problem. The management of the company cannot reach a final decision. But the data above will help the management make a decision once they all have the data needed (like the costs for expansion, etc.).