• No results found

The Queuing System of a Supermarket during the Corona Crisis

N/A
N/A
Protected

Academic year: 2021

Share "The Queuing System of a Supermarket during the Corona Crisis"

Copied!
53
0
0

Bezig met laden.... (Bekijk nu de volledige tekst)

Hele tekst

(1)

during the Corona Crisis

Submitted July 2020, in partial fulfillment of

the conditions for the award of the degree BSc Econometrics and Operations Research.

Fabienne S. Mouris

11804475

Supervised by mr Dr H.J. van der Sluis

Faculty of Economics and Business University of Amsterdam

(2)

Statement of Originality

This document is written by Student Fabienne Sarita Mouris who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of com-pletion of the work, not for the contents.

(3)

This paper illustrates the consumer behavior and experience in a supermarket by making a comparison of the period before and during the Corona Crisis. In the past Queuing Systems have been studied, but the impact of virus outbreak and intelligent lock down comes forward in this study. The data analysis finds that customers go less often to the supermarket during the Corona Crisis, but intent to buy more at the time. According to the steady state of an Open Queuing model with five stages, the shopping time of customers during the Corona Crisis is prolonged. This is in accordance with the Closed Network with the same number of stages. A contradiction with the Open Network is the customer direction that takes the longest which is shifted from the Checkout with personnel to the Scan&Go. The Closed Network model with three stages does coincide with the results of the network with five stages if the trolleys are followed. The restrictions implemented in the store do have an influence on the customer experience. By enlarging the arrival rate, the number of customers served converges to a limit below the arrival rate. By reducing the number of trolleys in the Closed Network the completion rate deduces and there is a fraction of customers who leave without doing groceries.

(4)

Acknowledgements

I gratefully thank my supervisor mr Dr H.J. van der Sluis who supported me throughout the entire process of writing a Bachelor Thesis in Operations Research. He suggested several ideas, but he also assisted me with setting up the models and solving mathematical errors. He provided supportive feedback and helped me with the small details. It was a pleasure to work with him.

I also acknowlegde Professor Dr N.M. van Dijk who contributed feedback on the first draft and helped me by understanding the product form of queuing networks and the proof that comes with it.

Thirdly, I would like to thank Alexander van Zadelhoff who provided me the data I used to analyze the models and all the correspondence regarding the restrictions in the supermarket. Until the last point he kept me in contact with Albert Heijn.

Lastly, I would like to thank family and friend who helped me to improve my final product by giving their opinions.

(5)

Statement of Originality ii Abstract iii Acknowledgements iv 1 Preface 1 1.1 Introduction . . . 1 1.2 Aims . . . 2 2 Theoretical Background 3 3 Data 8 4 Model 14 5 Results 19 6 Conclusion 27 7 Discussion 30 Bibliography 31 References 32 Appendices 34 A Figures 34 B Tables 36 v

(6)

C Formulas 38

D Proof of Product Form 41

(7)

4.1 Input parameters OQN Normal Period . . . 16

4.2 Input parameters OQN Corona Crisis . . . 16

4.3 Input Parameters CQN 5 stages Corona Crisis . . . 17

4.4 Input Parameters CQN 3 stages Corona Crisis . . . 18

5.1 Number of Customers and their time in the Supermarket in OQN . . . 21

5.2 Cr, Number of Trolleys and their time in the Supermarket in CQN . . . . 22

5.3 Closed Queuing Network with N=160 showing the customer results . . . . 25

(8)

List of Figures

3.1 Weekly Average Transactions per Hour . . . 10

3.2 Mondays Transactions . . . 10

3.3 Saturdays Transactions . . . 10

3.4 Mondays Revenue . . . 11

3.5 Saturdays Revenue . . . 11

3.6 Weekly Average Revenue per Hour . . . 11

3.7 Average Transactions per hour during the Normal Period . . . 12

3.8 Average Transactions per hour during the Corona Crisis . . . 12

4.1 Open Queuing Network Chart of a Supermarket . . . 18

4.2 Closed Queuing Network Chart of a Supermarket . . . 18

5.1 Cr with N = 16 fixed and increasing µ1 in 5 stages CQN . . . 24

5.2 Cr with µ1 = 19.7 fixed and increasing N in 5 stages CQN . . . 24

5.3 Cr with µ1 = 197 fixed and increasing N in 3 stages CQN . . . 25

(9)

Preface

1.1

Introduction

Waiting is something we experience all day. Basically, waiting is doing nothing. In fact, each room or area can become an waiting area. However, the physical context and the cultural expectations of the individual affect the experience of waiting. Standing in line for amusement is not the same as queuing up for supermarkets (Ehn & L¨ofgren, 2010). Since the outbreak of the COVID-19 - or Corona - virus, supermarkets are dealing with restrictions to minimize the spread of the virus, besides the national measures for every individual established by the government when announcing an intelligent lock down. It started with hygiene protection of employees (Pas, 2020) and later the government intro-duced a protocol to allow a maximum number of customers to enter the supermarket. In order to keep the 1.5 meter distance to one another, just 1 customer per 10 square meter is allowed. Therefore, the count of the number of trolleys or baskets limits the consumers in the supermarket (Rijksoverheid, 2020). Even a system is developed that counts the customers in the supermarket (Jansen, 2020). Before the outbreak of the Corona virus everyone could enter the supermarket without a trolley or basket. However, due to all restrictions, it might be that a queue originates before a customer can enter the supermar-ket. Consequently, all these changes influence the purchasing behavior of the customer.

(10)

2 Chapter 1. Preface

1.2

Aims

The changes the Corona virus has brought have a huge impact on daily life of everyone and even the most normal daily life activities cannot be executed the way people used to do. Therefore, I think, it is interesting to investigate the changed behavior of customers and the customer experience in supermarkets by comparing the situation before and during the Corona Crisis. In the beginning of the virus outbreak when the intelligent lock down has been introduced, customers went hoarding on food and other essential provisions (NOS, 2020). It might be that consumers go to the supermarket everyday since it is the only event of the day. On the other hand, they have more time to do all groceries at once and come less often than they used to. In combination with fear of getting infected they might avoid the supermarket as much as possible. Also, due to a restriction on the number of customers allowed in the supermarket, queues might establish.

In this paper the situation before the Corona Crisis will be compared with the situation during the Corona Crisis. The two periods will be distinguished by considering the situation before the Corona Crisis as ’Normal’. The Corona Crisis in the Netherlands started early March when the first contamination was registered and shortly after the first measures has been introduced.

The study starts with pointing out previous researches. With these insights two

models will be developed that present an open and a closed queuing network. Two

general models are shown which can be adjusted in such a way that it can be applied to most shops. I will present the model and analyze it based on a real data case. There is a chapter in which the data will be described. Moreover, from this data the behaviour of the consumer will be studied during the Normal and Corona period. The results of the model will present the consumer experience based on the queuing models developed and both situations, before and during the Corona Crisis, will be compared. Finally, the findings are summarized and concluded. The last section will bring improvements and further research forward.

(11)

Theoretical Background

In the 60’s Farley and Ring (1966) have done research on the traffic flow in super-markets. They developed a Markovian transition matrix as consumers decide amongst possibilities to move to another area in the store. The model formulation contained the design of the store and preference of sale for consumer as well as tendency for movement through the store areas. Eventually the probabilities are estimated by executing a re-gression of the independent variables on the probabilities. The research concludes that these independent variables rather have a positive and linear relation to the transition probabilities.

The cooperation of the operators of individual queues is investigated by Timmer and Scheinhardt (2010) to find out whether the operators can lower their costs assuming costs increase linear with the length of the queue. They found that two-node and three-node networks always have efficient cost allocation where both operators have an incentive. This study is elaborated to multiple node networks and again there always exists a cost allocation in which both operators have cost benefits even if costs aren’t based on queue lengths, waiting and service times (Timmer & Scheinhardt, 2013).

Despite the many studies done on and in supermarkets, the relevance to investigate supermarkets with respect to the waiting time nowadays has risen due to the outbreak of the COVID-19 virus. As already pointed out earlier, waiting for service is part of daily life. In fact, waiting costs time and time is money. By studying queues and the behavior of customers and servers, the waiting time can be reduced. When dealing with analysis of a queue, numerous concepts need to be considered. Customers arrive independently from each other, they will be served directly or add to the queue. However, the size of the

(12)

4 Chapter 2. Theoretical Background

queue plays a role and the selection procedure of the next customer too. Moreover, the queuing behavior of the customer should be analyzed. All along waiting, customers may change from a longer to a shorter queue, decide not to join a queue after arrival, or leave if they have to wait for too long. The service facility is another important aspect in the model. It should be considered whether there are one or more identical or non-identical servers available and what the flexibility of adding a server during rush hours is (Taha, 2017).

Queuing Network

A queuing model can be described shorthand by Kendall notations as A/S/c/K/Q where A describes the arrival process, S the service time, c the number of servers, K the capacity of the system and Q the queuing discipline. Designs to present A and S are several distributions such as Markovian (Exponential or Poisson), deterministic or general distribution. Typical queuing disciplines are first-come first-served (FCFS) and last-come first-served. If K and Q are omitted, then the capacity is unlimited and the queuing discipline is FCFS. All are single station queuing systems (Gass & Fu, 2013). However, in practice service systems may consist of multiple service stations. A supermarket consists of several stations where customers might have to wait. Service stations are the entrance, numerous areas within the store like the dairy department or the bakery and the check-out which is divided into different stations with various payment methods. A queuing network that has multiple stations and if each station has one queue, it is called a Jackson Network. Other assumptions for a network to be a Jackson Network is that the ith station has si servers and there is an unlimited waiting room at each station. In case of the

supermarket, each stage has its own number of servers dependent on the number of customers in the store. Moreover, there is enough space for customers to wait, ergo there is an unlimited waiting room. Further, customers arrive independently from each other at the ithstation from outside the network according to a Poisson distribution with intensity

λ and after customers are served with exponentially distributed service times they join the queue at station j with probability pi,j and leave the network with probability ri.

There exists a transition matrix which displays the probabilities to go from one stage to another (Kulkarni, 2011). Since the number of customers at each stage are independent variables, the system is called an Open Queuing Network, in short OQN (Winston &

(13)

Goldberg, 2004).

Generalized Jackson Networks that assume to have identically independent inter-arrival times and service times with general distributions are more real world systems (Garmarnik & Zeevi, 2006). The network has Bernoulli type routing and is focused on heavy traffic while proved that the stationary distribution of the Generalized Jackson Network converges to the stationary distribution of the associated Brownian motion.

The Supermarket Structure

A long time ago supermarkets were organized different than nowadays familiar. In the early 20thcentury supermarkets consisted of multiple full-service counters staffed by a number of servers such as a bakery, butcher or greengrocer, where customers were queuing up while waiting for service. Although supermarkets today consists of mainly self-service systems which significantly reduced the waiting time and consumer frustration, customers might have to wait to get goods from the shelves (Halper, 2008). Supermarkets can be seen as a Jackson Network if the customers arrive according to a Poisson process since they satisfy all other properties of a Jackson Network. However, it is a special application of the network. Customers always enter the network at the entrance of the supermar-ket, station 1. They can randomly do groceries through the supermarsupermar-ket, but they are mandatory to pay at the check-out counter before leaving the network. When the gather-ing goods part is considered as one station, that is all separate service stations in this part are neglected, the supermarket can be seen as a tandem queuing network. In this case, there are three service stations, one is the entrance, the second is doing groceries and the last one is checking out. The transition matrix takes a fixed form where the probabilities to go from one stage to the other are 1 (Kulkarni, 2011).

The check-out counter might be a simple station with identical servers. Yet, there are diverse ways to serve the customer. This last station in the queuing system of super-markets make use of lane selection (LS) models (Schwartz, 1974). These systems have various types of service facilities for various types of customers. Nowadays, the super-markets have several check-out possibilities. The most elaborated and service facilitated out counter accepts card and cash payments, the second type are self-service check-outs which only accept card payments and the express check-out for small orders only are the third type of service facility. In extremely modern supermarkets the customers

(14)

6 Chapter 2. Theoretical Background

have the opportunity to scan the products with your mobile phone or handscanner, so they only have to pay before leaving the store. Due to the separated payment methods, there are also three types of customers according the different service facilities. The first type is the customer who pays cash, the second type pays by card and the third type only purchases a small amount of articles or uses a handscanner and might use the fast lane. The property of the LS model is that customers of type i can be served by facility j for any value j ≤ i.

Closed Queuing Network

In a Jackson Network, all customers are arriving from outside the network and thus it is an Open Queuing Network. On the other hand, in a Closed Queuing Network (CQN) no arrivals or departures are permitted. In fact, there is a constant number of jobs cir-culating through the system. In a supermarket there is a fixed number of trolleys that customers can use. Since the beginning of the Corona Crisis, all customers are manda-tory to use a trolley. Therefore, the supermarket can be modeled as a CQN. Note that in the OQN the number of customers at each stage is independent, while in the CQN this is dependent as the number of trolleys in the system is fixed (Winston & Goldberg, 2004). Gordon and Newell (1967) developed an algorithm to determine steady-states in a CQN. There exists an equilibrium since there is assumed that the model has a stochastic structure of an irreducible, finite Markov process. In total there are N trolleys, divided over M stages and each stage has si parallel exponential servers. Also, the model possess

transition probabilities pij with i, j = 1, ..., M .

Product Form

The Open and Closed network are models to analyze the steady state distribution. When the steady state is factorized in structure to the stations, a product form is developed. According to van Dijk (1993) “a product form is defined as the factorization of the steady-state joint station distribution to the steady-steady-state single station distribution, up to nor-malization and its state space.” The use of the product form will validate the analysis of each station in a network separately. This product form can be characterized by notion of reversibility, that is if time would be changed, the system would emerge the same way.

(15)

The product form is constructed by equating the rate out of a state due to an exit with the rate into that state due to an arrival (van der Weij, van Dijk, & van der Mei, 2012).

Considering this background information, in this paper an Open and Closed Queuing Network Model is acquired. Before defining the models and its input parameters, a real data case is described. This data will be used in the models and to analyze the results.

(16)

Chapter 3

Data

For this research I obtained data from a supermarket of the company Ahold Del-haize, also called Albert Heijn (AH), established in the Netherlands. This is a rather large supermarket with opening hours daily from 8 am until 10 pm and employees working day and night to stock the shelves. The supermarket has one main entrance and outside next to the entrance the trolleys are received.. There is a common area customers go through before entering the store where goods can be gathered. From the moment the customer receives the trolley until the moment shopping starts will take 10 seconds. In the store, customers might move structured or randomly through the store but it is depending on which products they intent to buy. The store is divided in several departments and right after the entrance the vegetables and fruits can be gathered. This area is followed by the meat department and the bakery. Yet, customers may also skip these areas and go straight to the long-lasting products, like dried goods but also cosmetics. Another department, next to the fridges and long-lasting products, is an area for alcoholic beverages, sodas and chips. The last section before the check-outs contains deep-freezers. On average cus-tomers grab goods from the store in 15 minutes. Lastly, this supermarket has three types of paying their groceries at diverse spots located next to each other. The quickest way is by using the Handscanner. This is a portable scanner which is gathered at the entrance, but also a mobile phone with the ’appie app’ is applicable. When goods are grabbed, they can be scanned immediately and be put into the bag. At the exit the products only have to be paid for by card which only takes about 1 minute. The second option is to use the Scan&Go and the customer takes 3 minutes to scan and pay the goods. This Scan&Go area involve 14 self-checkout machines which only accepts card payments. The

(17)

most extended way to checkout is the regular Checkout. A cashier scans the products and card as well as cash payments are accepted. There are 5 check-out desks present, but if it is not necessary, that is not busy, only a few are in use. Most of the time the customer is served in 2 minutes. The trolleys are returned at its spot near the main entrance for the next customer.

Due to the outbreak of COVID-19, the supermarket was mandatory to adjust the store at some point to maintain the safety of employees and customers. Therefore, at the main entrance it is made sure every customers visits the store with a trolley which is cleaned by employees. From this moment it takes 25 seconds to enter the store as the trolley first needs to be cleaned. There is a restriction of the number of customers counted by providing 160 trolley. Moreover, there are only 10 Scan&Go machines available. Since at the checkout some hygiene precautions are introduced a cashier is capable to serve 25 customers per hour instead of 30 during the Normal period. Another modification are the opening hours. Since the beginning of the Corona Crisis the store opened an hour earlier, at 7 am, to serve exclusively elderly customers over the age of 70.

The data provided in for this research are the number of transactions and the corre-sponding revenue. The transactions have been made at a certain time moment, but this data set consists of the the number of transactions per hour. The revenue is the sum of the amount paid at each transaction in the according hour. The number of transactions per hour are split in the three possible payment methods. In order to compare two peri-ods, the data set contains transactions and revenues of weeks in the Normal period and Corona Crisis. These are respectively weeks 4, 5, 6 and weeks 13, 14, 16. The weeks do not enclose any holidays and the massive hoarding of customers during the first weeks of the Corona Crisis is excluded since that would have resulted in too many outliers.

Transactions and Revenues

First, the total number of transactions is analyzed in several ways. Figures 3.1, 3.2 and 3.3. clearly display a lower number of transactions in the weeks during the Corona Crisis than during the Normal period, except during the expanded opening hour for elderly customers. Noteworthy is the peak around lunch time and dinner time in the transactions on mondays during the Normal period, compared to the more continuous graphs during

(18)

10 Chapter 3. Data

the Corona Crisis. Since most of the labor is done from home, workers will not spend their break in the supermarket or pass by at the supermarket just before going home. Instead, customers have time during the day for a foray to the supermarket. In addition, on saturdays customers do their groceries more spread during the day in view of the more flat graphs. The hourly average of the 3 weeks coincide approximately and the busiest hours seems to be between 11 am and 7 pm.

Figure 3.1: Weekly Average Transactions per Hour

(19)

Figure 3.4: Mondays Revenue Figure 3.5: Saturdays Revenue

Figure 3.6: Weekly Average Revenue per Hour

The revenues on mondays, shown in figure 3.4, depict a higher revenue in the first week of the Corona Crisis. Also in the following weeks, the revenue is slightly higher apart from the evening hours. The revenue on saturdays did not increase in comparison to the Normal period, as a matter of fact it did not decrease considering the decline in the number of transactions. This can also be concluded from the average revenue displayed in figure 3.6. Overall, the revenue during the Corona Crisis increased as it is roughly the same in both periods, apart from the first hours when extra revenue is generated.

(20)

12 Chapter 3. Data

Arrival rate

Lets get back to establishing the arrival rate. Considering the average of the week is coinciding as concluded from the graphs in 3.3, it is not clear whether or not the struc-tures of the day are balanced. Hence, figure 3.7 and 3.8 present the average sales per hour of the day of the week in the Normal period and Corona Crisis respectively. There are clearly less transactions each day during the Corona Crisis than during the Normal pe-riod. Where the maximum number of transactions per hour in the Normal period is over 400, the maximum reached during the Corona Crisis is a slightly above 250 transactions per hour. Again, it is obvious that the peak before lunch- and dinner time in the Normal period has disappeared during the Corona Crisis. Nevertheless, the structure of the seven days are more or less similar in both periods. Therefore, the busiest hours of the day can be set from 11 am until 19 pm and these remain unchanged over time. All along these lines, the arrival rate per hour for every day of the week during the busiest hours will be the mean of the daily average. That has come to an average hourly arrival of 319 during the Normal period and 197 during the Corona Crisis.

Figure 3.7: Average Transactions per hour during the Normal Period

Figure 3.8: Average Transactions per hour during the Corona Crisis

Payment Probabilities

Since customers have 3 different possibilities to shop and pay but can only choose one, there needs to be determined what amount chooses a specific option. The same anal-ysis of the total transactions is applied to the payment methods. For the detailed graphs see figures A.1 until A.6 in the appendix where the data of the Handscanners, Scan&Go

(21)

and Checkout is displayed as average transactions per hour during a time period, Normal or Corona.

To start with the Handscanners, they are relatively new and therefore its use fluctu-ates more than the Scan&Go and checkout. Overall, the use of each payment method is equally spread over the days and the number of average transactions per day in relation to the average total transactions of a week in the corresponding period will give steady solutions. For the Normal period this will result in 17% of the customers will use the Handscanner, 46% pay at the Scan&Go and 37% prefer the Checkout with personnel. The percentages slightly differ during the Corona Crisis as 20% select the Handscanner to shop with, 48% will use the Scan&Go to pay and 32% of the customers choose the most extensive payment method.

(22)

Chapter 4

Model

In this part a model is given to investigate the consumer experience when doing there groceries in a supermarket before and during the Corona Crisis. The model is addressed with parameters of the real data case of AH. However, this model is suitable for a large store, but it can also be cut down to a small store. Therefore, the model presented here is not only relevant for the analyzed supermarket in this paper, yet other stores can make use of this model by changing the input parameters to the applicable parameters of its shop.

First, the model of the Open Queuing Network is acquired an a second model is given next considering a Closed Queuing Network. These models are established to determine the steady state situation of the number of customers in the supermarket. By applying Little’s Law the average time a customers spend in the supermarket is determined. To begin with, the Open and Closed Queuing Network consist of 5 stages which present the entire supermarket. So, the nodes of the queuing network are:

• Node (1) is the entrance. • Node (2) is the grocery section.

• Node (3) is the exit for customers who used a hand-scanner.

• Node (4) is the exit for customers who decided to queue up for the scan & go. • Node (5) is the exit for customers who pay at the check-out with personnel

assistance.

The second node, grocery section is taken as one stage although it actually exists of multiple areas where customers commute between. In this research the focus is on the

(23)

overall shopping experience and that is the reason to neglect the transitions of customers in the supermarket. Besides, there is no data available on the transition probabilities within the grocery section. By omitting this analysis the validity will rise since there is no need to assume on transition probabilities within the grocery section.

Let π(~n) denote the steady state distribution. The analysis of the Open Network is in principle based on M/M/s calculations per stage. Due to M/M/s calculations the steady state of one stage is determined independently of the other stages. This is only allowed on ground of the product form in equation 4.1. The steady state distribution in the Closed Network can’t be determined independently since the number of trolleys is fixed for the entire store and not per stage. Yet, a product form determines π(~n) shown in equation 4.2. A proof of the product form in the Open and Closed case can be found in Appendix D. π(n) = c 5 Y j=1 aj µj nj Y k=1 1 fj(k) = 5 Y j=1 πj(nj) (4.1) π(n) = c 5 Y j=1 λj µj nj Y k=1 1 fj(k) (4.2)

Open Queuing Network by Jackson

The supermarket is considered an Open Jackson Network with the following properties.

(1) The network has M=5 M/M/si nodes with i = 1, ..., M . Customers arrive from

outside the network at node i according to a Poisson distribution with intensity λi.

In the supermarket, customers can only arrive at the entrance, hence λ1 > 0 and

λ2 = ... = λ5 = 0.

(2) Node (i) has si identical servers working with an exponential distributed service rate

µi.

(3) When the customer is finished at node (i), with the probability pij the customer will

move to node (j) and leave the network with probability ri. Hence,

P5

j=1pij+ri = 1,

for all i = 1, ..., 5. (C.1)

(4) To ensure stability, ai < siµi where ai is the arrival intensity at node (i) including

the external and internal arrival intensity. That is, ai = λi+

P5

(24)

16 Chapter 4. Model

Figure 4.1 present the model where customers arrive from outside the system and within the pink dashed line customers are physically in the store. The analysis of this model is performed during the Normal and Corona period. An overview of the input parameters are given in table 4.1 and 4.2. The inter-arrival rate a is obtained by matrix multiplication involving the identity and P matrix and the external arrival vector λ (C.2). The number of servers and its service times are elucidated in the previous section, data. Meaningful are the number of servers at stage 2 and 3. There were infinite servers during the Normal period as every customers could walk into the supermarket and serve themselves. However, the restriction of 160 in the entire system caused a reduction in stage 2 and 3 below 160. This reduction will not change the results when the number of servers in stage 2 and 3 is 160, but it justifies the total of 160 customers in the store.

Normal Stage 1 Stage 2 Stage 3 Stage 4 Stage 5

a 319 319 53 147 119

s 1 ∞ ∞ 14 5

µ 360 4 60 20 30

Table 4.1: Input parameters OQN Normal Period

Corona Stage 1 Stage 2 Stage 3 Stage 4 Stage 5

a 197 197 38 95 64

s 1 115 50 10 3

µ 240 4 60 20 25

Table 4.2: Input parameters OQN Corona Crisis

Closed Queuing Network by Gordon and Newell

The supermarket is considered a Closed Queuing Network where the trolleys are followed and will not leave the system. The network has the following properties.

(1) The network has M = 5 stages where stage i for i = 1, ..., M has ni trolleys, either in

service or in the queue, such that PM

i=1ni = N . Customers arrive from outside the

network at the entrance, although this is not allowed in a Closed Network. Hence the service rate at the first stage is translated into a arrival rate and λ defines the way the trolley traverse. There is no transit time between the stages.

(2) Node (i) has si identical servers working with an exponential distributed service rate

(25)

(3) When the customer is finished at node (i), with the probability pij the customer

will move to node (j). Hence, P5

j=1pij = 1, for all i = 1, ..., 5. (C.1)

(4) The number of observable state combinations is given by N and M , that is   N + M − 1 N  .

The model of the Closed Network is presented in figure 4.2. The trolleys are the entire time in the system and once leaving one of the payment stages it commutes back to the entrance where the trolleys waits to circulate through the store again. The model is only inspected during the Corona Crisis as this model is not representative during the Normal period when there is no restriction on the number of customers entering the store. The input parameters slightly differ from the Open Network. Instead of inter-arrival rates, this model presents the customer flow by λ. That is every customer who enters the store, will shop at the second stage and pays with a certain probability at stage 3, 4 or 5. Moreover, the service rate at the entrance is the arrival rate of customers. This is an approach to implement the number of transactions made per hour into the model of the Closed Network. Although the arrival rate was 197 per hour before, the service rate at the entrance is 19.7 per hour. As consequent of a quickly increase in the number of combinations by raising the fixed number of trolleys in the system (A.7), it is decided to analyzed the model with N=16 and M=5 and 4845 combinations. Therefore, only 101 of the ordinary input is taken in terms of the servers, number of customers arriving and the maximum amount of trolleys in the store. The number of servers are rounded up to natural numbers.

Since the number of combinations reduces by deducing the number of stages given the number of trolleys, it is possible to analyze the Closed Network for a higher number of trolleys. Therefore, the Closed Network is analyzed with 3 stages. Each payment method is simulated in the third stage once with its own input parameters as defined in table 4.4.

Corona Stage 1 Stage 2 Stage 3 Stage 4 Stage 5

λ 1 1 0.1949 0.4824 0.3226

s 1 16 16 1 1

µ 19.7 4 60 20 25

(26)

18 Chapter 4. Model

Corona Stage 1 Stage 2 Stage 3 Stage 4 Stage 5

λ 1 1 0.1949 0.4824 0.3226

s 1 80 60 10 3

µ 197 4 60 20 25

Table 4.4: Input Parameters CQN 3 stages Corona Crisis

Figure 4.1: Open Queuing Network Chart of a Supermarket

(27)

Results

In this section the results are discussed. MS Excel is used to model the Open Queuing Network and the Closed Queuing Network.

AH case OQN

First, the results of the Open Queuing Network by Jackson are discussed. The network has 5 stages where each stage will be analyzed separately as an M/M/s system. The number of servers and its service time is defined in table 4.1 and 4.2. Important are ρi

and r (C.3), the utilization of the servers and the number of customers at the server. If ρ is smaller than 1 the system is stable and a steady state will be the result. These parameters will be used in the calculations of the M/M/s stages. For s equal to one, finite or infinite, the probability of an idle stage is determined first as the probability that n customers are in the stage depends on the probability that zero customers are in the stage. It can also be analyzed as a recursive formula depending on the probability that n − 1 customers are in the stage. The formulas for calculating the probability that n customers are in the system are different for some s. There are three cases to distinguish with 1 server, finite servers or infinity many servers (C.4, C.5, C.6). Once the probabilities regarding the number of customers are found, the steady state for the number of customers in the system (L) can be determined as well as the number of customers in the queue (Lq). The average

time customers spend in the system can be derived by applying Little’s Law (C.7), that is dividing the number of customers in the system by its arrival rate. This can be done for the total number of customer in the system as well as the number of customer per stage and taking its inter-arrival rate. Adding up the customers at the server and in the

(28)

20 Chapter 5. Results

queue will result in the total customers in the system and doing this with the waiting and serving time will result in the total time in the system (C.8).

The steady state on the number of customers and their time spent in the supermarket is presented in table 5.1 for the Normal period and the Corona Crisis. A more detailed output is shown in Table B.1 and B.2 of Appendix B. The times (W) are in minutes. The last column ’Total’ describes the average results of a customer who visit the supermarket. A straightforward consequence of the 38% decrease in number of transactions per hour since the beginning of the Corona Crisis is the decrease in the average number of customers in the system of 34%. The slightly smaller deduction in the number of customers have been caused by less servers at the Scan&Go and the Checkout. Also the longer processing time at the Checkout contributes. Accordingly the number of customers in the queue together with the waiting time increased at these two stages. The prolonged service time at the Entrance does not have significant impact. In fact, the total time at the Entrance has decreased since there is a reduced waiting time for entering the supermarket during the Corona Crisis compared to the Normal period, only 1.16 minute in Corona Crisis as 1.31 minute during Normal period. In other words, there are less customers waiting at the Entrance to get into the supermarket in contrast with the number of customers in the queue during Normal times. Overall, the number customers in the queue have been diminished from 9.57 to 8.69. However, the total average time any customer spends in the supermarket slightly increased with 6.05% from 19.15 to 20.31 minutes, which involves both the waiting time and the service time. This can be explained by looking at the average time of a customer direction (B.3, B.4). The total average (W) can also be calculated based on the percentages of customers choosing the payment method. It can be concluded that the waiting time for the checkout direction as well as the service time has been increased during the Corona Crisis to 23.36 minutes in total compared to the Normal period with a time of 19.77 minutes. Hence, the total average time has been enlarged. In fact, the expansion of the total average time to 20.31 minute does not do justice to a specific customer, by reason of customers choosing the Handscanner or the Scan&Go as payment method spend slightly less time in the supermarket while customers who pay at the checkout spend more time.

In the above analysis and as a characteristic of an M/M/s calculation, all stages enclose just one queue. For the first four stages it is relatable to the reality. However, the

(29)

last stage, Checkout, has in reality the number of queues as the amount of servers avail-able. In other words, each Checkout-desk has their own queue. Nevertheless, performing M/M/1 calculations given 1s customers arrive at a queue does not reflect reality. These queues do not allow jockeying resulting in exceptional waiting times, which resulted in the use of only 1 queue.

OQN Entrance Shopping Handscan Scan&GO Checkout Total

Normal Period L 7.84 79.82 0.89 7.39 5.99 101.93 W 1.47 15.00 1.00 3.01 3.03 19.15 Corona Crisis L 4.63 49.35 0.64 4.81 7.38 66.82 W 1.41 15.00 1.00 3.03 6.95 20.31

Table 5.1: Number of Customers and their time in the Supermarket in OQN

AH case CQN

The second model, the Closed Queuing Network, is only analyzed during the Corona Crisis. In a Closed Network there is a fixed number of customers with trolleys circulat-ing through the system which is a contradiction to the Open Network where customers come and go. Since the Closed Network cannot be analyzed by looking at each stage separately, all combinations from point 4 of the closed network model and its probability are determined which make partially use of the M/M/s calculations (C.9) and uses an approach function to derive the probability that stage i contains ni trolleys. This

equa-tion together with the occupancy rate will lead to the probability that the stage is busy. The completion rate (C.10) is based on the probability that the stage is busy multiplied by the service rate and the number of servers. More important, with all this information the steady state distribution of the number of trolleys is determined to show that there are indeed N trolleys in the system divided of M stages. The number of trolleys in the system and in the queue per stage is determined the same way as the M/M/s calculations in the Open Network. The waiting time can still be derived according Little’s Law, but the completion rate, also λef f ective, should replace λ.

There are several ways the Closed Model is analyzed. Firstly, the model with 5 stages is investigated. From the data it follows that there are 160 customers accepted in

(30)

22 Chapter 5. Results

the supermarket at a certain moment during the Corona Crisis. However, this appears to be a tremendous number of combinations, since the number of combinations increases sharply compared to the increase of N (A.17). By a number of 20 customers, there are already 10626 feasible combinations while at N=16 this is 4845 (A.18). Hence, the input parameters from the real data are scaled to an allowance of 16 customers in the supermar-ket (B.15). The input of λ determine the distribution over the network. Let one customer enter, then the customer will shop twice and pay partially at one of the payment methods. The service rate at the first stage is transformed to the arrival rate based on the real data.

CQN Entrance Shopping Handscan Scan&GO Checkout Total

Cr 19.67 19.67 3.84 9.49 6.35 19.67

L 9.78 4.92 0.06 0.90 0.34 16.00

W 29.82 15.00 1.00 5.68 3.21 48.79

Table 5.2: Cr, Number of Trolleys and their time in the Supermarket in CQN

The steady state distribution of the division of the trolleys is shown in table 5.2 together with the completion rate per stage and the time the trolley spends at each stage. The last column shows the total completion rate of the entire system per hour and the time it takes for a trolley to go around the entire store. The total trolleys remains 16 since there are no arrivals or exits. The completion rate of 19.67 is the effective arrival rate which implies that 0.03 customers who arrive immediately leave and do not enter the supermarket. As this number is little, it can be assumed that all customers who arrive, will be served. The steady state for the number of customers in each stage is equal to the number of trolleys, except at the entrance. Here, the trolleys wait for a customer to enter the supermarket. Hence, it is reasonable to speculate that customers do not have to wait before entering the supermarket. The total average of customers in the supermarket is given by the total customers at stage 2 until 5 including the customer at the server in stage 1, which will be 7.22. The total time a customer spends in the supermarket is 19.69 minutes on average. Due to the probabilities corresponding with the payment method, the Handscanner is the quickest way to pay with a total duration of 16.42 minutes while the Scan&Go takes the longest 21.28 minutes. These shopping times differ from the outcome of an Open Network during the Corona Crisis. Instead of Scan&Go, in the the Checkout with personnel takes the longest and the total average shopping time compared to the

(31)

Normal period is slightly higher with 0.43 minute, or 26 seconds.

The results of the closed network with 5 stages is analyzed, but these are ratios. It is moreover interesting to look at the effects of changing the number customers allowed in the supermarket as well as changes in the arrival rate. In the Closed Network with 5 stages analyzed with the data from the store is found that all customers who arrive are served, that is the average arrival rate of 197 customers per hour by an allowance of 160 customers in the supermarket at the time. Figure 5.1 displays the effects of the completion rate, or effective arrival, when the arrival rate translated to µ1 is enlarged.

At all times the completion rate is below the increasing arrival rate µ1. As the arrival

rate expands, the completion rate is coming withal to a steady level, in other words the completion rate converges to a value which is beneath the arrival rate. This means that not all arrived customers are being served, instead there is a number of customers who immediately leave. On the other hand, let µ1 = 19.7 be constant and the number of

customers permitted in the supermarket increasing. Figure 5.2 shows that for N small, the completion rate is below the arrival rate. From approximately N=10 the completion rate will approach the arrival rate. Once the completion rate reaches the arrival rate, it has reached its steady state. Henceforth, all customers who arrive will be served and none is pulling out.

(32)

24 Chapter 5. Results

Figure 5.1: Cr with N = 16 fixed and increasing µ1 in 5 stages CQN

(33)

For further analysis the Closed Network is reduced to 3 stages. The weighted average of all payment methods following the trolleys in the system is compared with the outcome of the Closed Network with 5 stages to investigate the validity of the ratios used. The average time a trolley takes to visit the store once is 48.73 minutes which is only 3.6 seconds less than the result in the Closed Network with 5 stages. For the detailed information see table B.5 in Appendix B. The customer results are shown in table 5.3 as weighted average in the column total and per customer direction. Meaningful is the average shopping time at 18.90 minutes with a small fraction of 1.24 minutes waiting time. This waiting time is originated at the Checkout with personnel. The Handscanner and Scan%Go barely have any waiting time before being served. This is in contradiction with the outcome of the Closed Network with 5 stages, but is more in line with the outcome of the Open Network during the Corona Crisis.

Direction Total Handscan Scan&Go Checkout

L 54.80 50.89 55.03 56.82

W 18.90 16.25 18.27 21.46

Wq 1.24 0.00 0.02 3.81

Ws 17.67 16.25 18.25 17.65

Table 5.3: Closed Queuing Network with N=160 showing the customer results

Figure 5.3: Cr with µ1 = 197 fixed and increasing N in 3 stages

(34)

26 Chapter 5. Results

It is interesting to analyze the restrictions in the supermarket as a consequence of the Corona virus related measurements. Therefore, let the hourly arrival rate translated to µ be fixed at 197 while N is increasing. Since the difference in the Completion Rate between the payment negligible, the Scan&Go payment method is presented in figure 5.3 and the Handscanner and Checkout are omitted. So, the graph demonstrates that the completion rate ascents by enlarging N. From approximately N=80, the completion rate is equal to the hourly arrival rate. From that point, all arrived customers will be served. In fact, although there are 160 servers in the second stage of the system, 80 servers would be enough to obtain the same results as long as the number of trolleys is 160. When N is low, there barely is any waiting time, because as soon as a trolley finishes its round in the supermarket, a new customer has been arrived. However, the leaving rate of customers not being served is also significant since the completion rate is below the arrival rate.

(35)

Conclusion

In this paper I compared the customer behavior and experience in a supermarket before and during the Corona Crisis. An Open and Closed model is developed and the results of these models are based on a real data case. Both networks consist of 5 stages displaying the entrance, the grocery area and three different payment methods, the Handscanner, Scan&Go and the Checkout with personnel. The product form justifies the independent analysis of each stage in the steady state distribution.

The consumer behavior is based on the data analysis by looking at the number of transactions per hour and the according revenue. The busiest hours during the day didn’t alter since the outbreak of the Corona virus. The most evident change in the behavior of the customer due to the outbreak of COVID-19 and the introduction of the intelli-gent lock down is the number of visits to the supermarket which decreased tremendously. Moreover, customers come more spread throughout the day. As a consequence the peaks during lunch and before dinner time have disappeared. Although a visit to the super-market may be the only activity during the day, customers do have time to do weekly groceries at once and might avoid the supermarket in fear of the virus. Customers doing weekly groceries is confirmed by looking at an increased revenue per transaction. The probabilities for the payment methods slightly changed, but will not have a vast impact on the difference in the steady state distribution of the Normal and Corona period. These probabilities for choosing one of the three payment methods determine the customer flow in the store, that is the number of customer that enters a certain stage. The steady state distribution of the Open Network is obtained via M/M/s calculations and show that there

(36)

28 Chapter 6. Conclusion

are less customers in the store during the Normal period than in the Corona Crisis. This is in line with a lower arrival rate although the capacity of the store has been diminished. The total shopping time increased with 1.10 minutes which is caused by the increased waiting and service time at the Checkout. The visit of customers paying at the Checkout with personnel takes the longest, while paying by Handscanner is evidently the shortest visit to the store.

The Closed Network is only applicable to the Corona time period since there is a fixed number of trolleys available in the store without arrivals and departures, where in the Normal period customers entered the store also with a basket or empty handed. The analysis consists of a Closed Network with 5 stages and 16 trolleys in the store and an-other with 3 stages and 160 trolleys due to enormous increase in the number of feasible state combinations by enlarging the number of trolleys or stages. The 5 stages network calculates the steady state in ratios of the real data case with the result that almost every customer arrived customer is served. The most important finding is an average shopping time of 19.69 minutes which is somewhat smaller than the shopping time in the Open Network of 20.31 minutes. In contradiction to the Open Network, it is not the Checkout which is the most time consuming shopping direction, but the Scan&Go. This might be the result of rounding the number of servers to natural numbers. The number of servers at the Scan&Go is an exact ratio, but not at the Checkout where the number of servers is actually smaller than one by taking the exact ratio. The network with 3 stages is analyzed 3 times, each with the different payment method at the third stage. This re-sults in a lower average shopping time than in the Normal period. However, the Checkout has the longest shopping time which is in accordance with the results of the Open Network.

Another interesting inquiry of the Closed Network is setting the parameters included the number of trolleys to a specific value except the service rate, which is actually the arrival rate, at the first stage. Although the completion rate converges to a constant rate when the arrival rate is increasing, this completion rate is very much below the arrival rate and thus many customers leave immediately before entering the store. If the arrival rate is fixed and let the number of trolleys be changing, then in the system with 16 trolleys it is found that from approximately 10 trolleys every arriving customer is served, while this is 80 trolleys in the closed network with 3 stages and 160 trolleys available.

(37)

Overall, it can be found that shopping during the Corona period is less busy with less customers in the store at the time but it takes a bit longer due to measures taken by the government. In fact, the restriction on the number of trolleys in the store does influence the total number customers served per hour in a decreasing manner.

(38)

Chapter 7

Discussion

This section elaborates on some of the pitfalls and shortcomings of this investigation. Besides, it contains suggestions for further research on Queuing Systems in Supermarkets.

First of all, all calculations are based on an average arrival rate amongst the busiest hours during the day, similar with the percentages of choosing a payment method and the number of servers available. A remark, it depicts only a small fraction of the real world. In further researches a simulation of the situation is able to investigate the customer ex-perience more precisely and elaborated over the day, instead of only during the busiest hours of the day.

The probability to find n customers in a stage are numerically calculated in the entire paper. This might cause issues for large n as excel determines the probabilities for each cell from scratch. Therefore, recursive formulas improve pace of the calculations and excel is most likely to process a higher number of calculations. The outcome is identical to the numerical outcome when applying the suitable recursive formula. There are also other programming languages like Matlab or R which are appropriate for the calculations of high order and are competent to more sophisticated circumstances.

This research paper makes use of standard M/M/s computations with each stage only having one queue regardless the number of servers. In reality this is not the case for each stage, like the Scan&Go have more or less 1 or 2 queues and the Checkout has a queue for each Checkout desk. The arrivals of the Checkout stage could have been evenly split over each counter, but this results in surrealistic queue lengths and waiting time. Best would be to have multiple queues allowing jockeying and choosing the shortest

(39)

queue. For this paper I decided to base the results on one queue for all server-desks, but to purify the outcomes the model should be adjusted. I mainly analyzed the steady state distribution. Nevertheless, there is more potency in the field of studying the distributions of the probability that n customers are in the system or n trolleys at a certain stage.

(40)

References

Ehn, B., & L¨ofgren, O. (2010). The secret of doing nothing: waiting. University of California Press.

Farley, J. U., & Ring, L. W. (1966). A Stochastic Model of Supermarket Traffic Flow. Operations Research, 14 (4), 555-567. doi: https://doi.org/10.1287/opre.14.4.555 Garmarnik, D., & Zeevi, A. (2006). Validity of Heavy Traffic Steady-Stat

Approxima-tions in Generalized Jackson Networks. OperaApproxima-tions ResearchThe Annals of Applied Probability, 16 (1), 56-90. doi: https://doi.org/10.1214/105051605000000638

Gass, S. O., & Fu, M. C. (2013). Kendall’s Notation. Enceyclopedia of Operations Research and Management Science, 13 . doi: https://doi.org/10.1007/978-1-4419-1153-7 200360

Gordon, W. J., & Newell, G. F. (1967). Closed Queuing System with Exponential Servers. Operations Research, 15 (2), 254-265. doi: https://doi.org/10.1287/opre.15.2.254 Halper, E. B. (2008). Supermarket Use and Exclusive Clauses, Part Six. Real Property,

Trust and Estate Law Journal , 43 (2), 224-309.

Jansen, R. (2020). Ah franeker telt klanen met slimme camera. doi:

Retrieved 22.04.2020, https://www.distrifood.nl/ondernemen/nieuws/2020/03/ah-franeker-telt-klanten-met-slimme-camera-101133272

Kulkarni, V. (2011). Introduction to Modeling and Analysis of Stochastic Systems, 2nd edition. In (chap. 6 Queueing Systems). Springer Science + Business Media. NOS. (2020). Supermarkten: ’alle hens aan dek, maar hamsteren is echt niet nodig’.

doi: Retrieved 22.04.2020, https://nos.nl/artikel/2326980-supermarkten-alle-hens-aan-dek-maar-hamsteren-is-echt-niet-nodig.html

Pas, H. t. (2020). Corona: Ah schermt caissi`e af met tafelzeil. doi: Retrieved

22.04.2020,

https://www.distrifood.nl/ondernemen/nieuws/2020/03/corona-ah-schermt-caissiere-af-met-tafelzeil-101132703 32

(41)

Rijksoverheid. (2020). Letterlijke tekst persconferentie minister-president rutte, minis-ters grapperhaus, de jonge en van rijn over aangescherpte maatregelen coronavirus. doi: Retrieved 22.04.2020, https://nos.nl/artikel/2326980-supermarkten-alle-hens-aan-dek-maar-hamsteren-is-echt-niet-nodig.html

Schwartz, B. L. (1974). Queuing models with Lane Selection: A New Class of Problems. Operations Research, 22 (2), 331-339. doi: https://doi.org/10.1287/opre.22.2.331 Taha, H. A. (2017). Operations Research An Introduction. In (chap. 18: Queuing

Sys-tems). Pearson Education Limited.

Timmer, J. B., & Scheinhardt, W. R. W. (2010). How to share the cost of cooperating queues in a tandem network? Proceedings of the 22nd International Teletraffic Congress (ITC), (ITC), 1-7. doi: https://doi.org/10.1109/ITC.2010.5608712

Timmer, J. B., & Scheinhardt, W. R. W. (2013). Cost sharing of

cooper-ating queues in a Jackson Network. Queuing Systems, 75 (1), 1-17. doi:

https://doi.org/10.1007/s11134-012-9336-4

van der Weij, W., van Dijk, N., & van der Mei, R. (2012). Product form results for two station networks with shared resources. Performance Evaluation, 69 (12), 662-683. doi: https://doi.org/10.1016/j.peva.2012.08.002

van Dijk, N. M. (1993). Queueing networks and product forms: a systems approach (Vol. 4). John Wiley & Son Limited.

Winston, W. L., & Goldberg, J. B. (2004). Operations research applications and algo-rithms. Thomson Learning.

(42)

Appendix A

Figures

Figure A.1: Average Transactions

Handscanners per hour during the Nor-mal period

Figure A.2: Average Transactions

Handscanners per hour during the

Corona Crisis

Figure A.3: Average Transactions

Scan&Go per hour during the Normal period

Figure A.4: Average Transactions

Scan&Go per hour during the Corona Crisis

(43)

Figure A.5: Average Transactions Checkout per hour during the Normal period

Figure A.6: Average Transactions

Checkout per hour during the Corona Crisis

(44)

Appendix B

Tables

Normal Entrance Shopping Handscan Scan&GO Checkout Total

L 7.84 79.82 0.89 7.41 6.52 102.48 Lq 6.96 0.00 0.00 0.04 2.57 9.57 Ls 0.89 79.82 0.89 7.37 3.95 92.92 W 1.47 7.50 1.00 3.02 3.30 19.26 Wq 1.31 0.00 0.00 0.02 1.30 1.80 Ws 0.17 7.50 1.00 3.00 2.00 17.46

Table B.1: Open Jackson Queuing Network results per stage during the Normal period

Corona Entrance Shopping Handscan Scan&GO Checkout Total

L 4.63 49.35 0.64 4.81 7.38 66.82 Lq 3.81 0.00 0.00 0.05 4.83 8.69 Ls 0.82 49.35 0.64 4.76 2.55 58.12 W 1.41 7.50 1.00 3.03 6.95 20.31 Wq 1.16 0.00 0.00 0.03 4.55 2.64 Ws 0.25 7.50 1.00 3.00 2.40 17.67

Table B.2: Open Jackson Queuing Network results per stage during the Corona Crisis

Customer direction Handscan Scan&GO Checkout

W 17.47 19.49 19.77

Wq 1.31 1.33 2.61

Ws 16.17 18.17 17.17

Table B.3: Open Jackson Queuing Network waiting and service times per customer-direction during the Normal period

(45)

Customer direction Handscan Scan&GO Checkout

W 17.41 19.44 23.36

Wq 1.16 1.19 5.71

Ws 16.25 18.25 17.65

Table B.4: Open Jackson Queuing Network waiting and service times per customer-direction during the Corona Crisis

CQN Entrance Shopping Handscan Scan&GO Checkout Total

Cr 19.67 19.67 3.84 9.49 6.35 19.67 L 9.78 4.92 0.06 0.90 0.34 16.00 Lq 8.78 0.00 0.00 0.42 0.09 9.29 Ls 1.00 4.92 0.06 0.47 0.25 6.71 W 29.82 15.00 1.00 5.68 3.21 48.79 Wq 26.78 0.00 0.00 2.68 0.81 28.33 Ws 3.05 15.00 1.00 3.00 2.40 20.46

Table B.5: Closed Queuing Network with N=16 showing the trolley results

Customer direction Total Handscan Scan&GO Checkout

L 7.22

W 19.69 16.42 21.28 18.76

Wq 1.81 0.00 2.86 0.94

Ws 17.67 16.42 18.42 17.82

Table B.6: Closed Queuing Network per Customer Direction

Direction Total Handscan Scan&Go Checkout

Cr 197.00 197.00 197.00 197.00 L 160.00 160.00 160.00 160.00 Lq 106.51 109.11 105.00 107.21 Ls 53.49 50.89 55.00 52.79 W 48.73 48.73 48.73 48.73 Wq 32.44 33.23 31.98 32.65 Ws 16.29 15.50 16.75 16.08

(46)

Appendix C

Formulas

P =       0 1 0 0 0 0 0 p23 p24 p25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0       , R =0 0 1 1 1 (C.1)

Equation C.1: Probability Transition (P) and Leaving (R) Matrix

λ = λ1 0 0 0 0 , a = (I − P )−1λ (C.2)

Equation C.2: External arrival rate (λ) and Inter-arrival rate (a)

ρi = ai siµi , ri = ai µi (C.3)

Equation C.3: ρ and r of stage i

(47)

p0 = 1 − ρ pn= ρnp0 L = ρ 1 − ρ Lq= ρ2 1 − ρ (C.4) Equation C.4: M/M/1 calculations p0 = s−1 X n=0 rn n! + ∞ X n=s rn s!sn−s !−1 pn = ( rn n!p0 if n < s rn s!sn−sp0 if n ≥ s L = ∞ X n=0 npn Lq = ∞ X n=s (n − s)pn (C.5) Equation C.5: M/M/s calculations p0 = ∞ X n=0 rn· exp−r n!  !−1 pn= rn· exp  −r n!  p0 L = r (C.6) Equation C.6: M/M/∞ calculations L = λW (C.7)

(48)

40 Chapter C. Formulas L = Ls+ Lq, Ls = r W = Ws+ Wq, Ws = 1 µ (C.8)

Equation C.8: Customers and Time in System, at Server, in Queue

pi,n=

X

xi=ni,j

pj(N, M ) with xi the number of trolleys at stage i

and pj(N, M ) = QM i=1zj(ni,j) G(N, M ) , zj(ni) =    rnii ni! if ni < si rnii si!sni−sii if ni ≥ si , G(N, M ) =X j M Y i=1 zj(ni,j) (C.9)

Equation C.9: The probability that there are n customers at stage i

Cr =Pbusy· µ · s Pbusy = N X n=0 min(s, n) s · pn (C.10)

(49)

Proof of Product Form

Open Product Form

The product from for the Open Queuing Network is:

π(n) = c 5 Y j=1 aj µj nj Y k=1 1 fj(k) = 5 Y j=1 πj(nj) (D.1) with, 5 Y j=1 1 fj(k) =        1 nj! if nj < sj 1 sj!snj −sjj if nj ≥ sj

In order to prove that D.1 holds, the global balance equation should be verified in such a way that the inflow and outflow are equal in each stage. Therefore, there is an indicator function introduced. 1(condition) =      1 if condition is true 0 if condition is false 41

(50)

42 Chapter D. Proof of Product Form

The global balance equation is:

(2.1) π(~n)f1(n1)µ11(n1>0)+ (2.2) π(~n)f2(n2)µ21(n2>0)+ (2.3) π(~n)f3(n3)µ31(n3>0)+ (2.4) π(~n)f4(n4)µ41(n4>0)+ (2.5) π(~n)f5(n5)µ51(n5>0)+ (2.6) π(~n)λ1P5 j=1nj<T )                            =                            π(~n − e1)λ1(n1>0)+ (2.1’) π(~n + e1− e2)f1(n1+ 1)µ11(n2>0)+ (2.2’) π(~n + e2− e3)f2(n2 + 1)µ2p231(n3>0)+ (2.3’) π(~n + e2− e4)f2(n2 + 1)µ2p241(n4>0)+ (2.4’) π(~n + e2− e5)f2(n2 + 1)µ2p251(n5>0)+ (2.5’) P5 j=3π(~n + ej)fj(nj+ 1)µjpj01(P5 j=1nj<T ) (2.6’) (D.2)

In more detail, the D.1 can be proven for each equation separately. That will result in balance equations per stage, 6 in total. That is including a station for the external world. It will be shown that (2.i) = (2.i’) for i = 1, ..., 6. Each equation has an interpretation of physical inflow and outflow of a station. First note that the indicator function is the same on both sides of equations (2.i) = (2.i’). In general, for each station j the equation to be solved is: (a) π(n)fj(nj)µj = 5 X i=1 π(n + ei− ej)fi(ni+ 1)µipij + π(n − ej)λj (b) π(n) 5 X j=1 λj = 5 X j=3 π(n + ej)fj(nj + 1)µjpj0 (D.3)

Since the indicator function is the same on both sides of the equation of D.3, it will cancel out. From D.1 there is directly concluded that for each equation

π(n + ei− ej) π(n) =  ai µi   µj aj  fj(nj) fi(ni+ 1) π(n + ei) π(n) =  ai µi  1 fi(ni+ 1) π(n − ej) π(n) =  µj aj  fj(nj) 1

(51)

Next, these ratios can be substituted into the each of the equations (2.i) = (2.i’) of D.2 which results in:

a1 = λ a2 = a1 a3 = p23a2 a4 = p24a2 a5 = p25a2 λ = a2+ a3 + a4

Altogether, this results in the traffic equation D.5 which completes the proof.

aj =

X

i

aipij + λj (D.4)

Closed Product Form

The product from for the Closed Queuing Network is:

π(n) = c 5 Y j=1 λj µj nj Y k=1 1 fj(k) (D.5)

where fj(nj) remains unchanged.

In order to prove that D.5 holds, the global balance equation should be verified in such a way that the inflow and outflow are equal in each stage. The proof is similar to the Open Case only the formula differs slightly, namely in the first row and the last row is omitted.

(52)

44 Chapter D. Proof of Product Form

The global balance equation is:

(6.1) π(~n)f1(n1)µ11(n1>0)+ (6.2) π(~n)f2(n2)µ21(n2>0)+ (6.3) π(~n)f3(n3)µ31(n3>0)+ (6.4) π(~n)f4(n4)µ41(n4>0)+ (6.5) π(~n)f5(n5)µ51(n5>0)                      =                      P5 i=3π(~n + ei− e1)fi(ni+ 1)µi1(n1>0) (6.1’) π(~n + e1− e2)f1(n1+ 1)µ11(n2>0)+ (6.2’) π(~n + e2− e3)f2(n2+ 1)µ2p231(n3>0)+ (6.3’) π(~n + e2− e4)f2(n2+ 1)µ2p241(n4>0)+ (6.4’) π(~n + e2− e5)f2(n2+ 1)µ2p251(n5>0)+ (6.5’) (D.6)

Again, D.5 can be proven for each row separately resulting in (6.i)=(6.i’) for i = 1, ..., 5 presenting the physical outflow and inflow per station. In general, the equation to be solved for each station j is:

π(n)fj(nj)µj = 5

X

i=1

π(n + ei− ej)fi(ni + 1)µipij (D.7)

From D.6 it directly follows that for each equation:

π(n + ei− ej) π(n) =  λi µi   µj λj  fj(nj) fi(ni+ 1) π(n + ei) π(n) =  λi µi  1 fi(ni+ 1) π(n − ej) π(n) =  µj λj  fj(nj) 1

Next, these ratios can be substituted into each of the equations (6.i)=(6.i’) of D.6 which results in:

(53)

a1 = a3+ a4 + a5

a2 = a1

a3 = p23a2

a4 = p24a2

a5 = p25a2

Altogether this results in the traffic equation D.8 which completes the proof. Since this is the proof of the closed network, there is a degree of freedom in the traffic equation. However, this is corrected by the fixed number of trolleys in the system.

aj =

X

i

Referenties

GERELATEERDE DOCUMENTEN

In the third chapter, the European visa policies regarding to non-EU students, the existence of non European member states in the EU, the mobility of non-EU students in the

(2010) Phishing is a scam to steal valuable information by sending out fake emails, or spam, written to appear as if they have been sent by banks or other reputable organizations

(2010) Phishing is a scam to steal valuable information by sending out fake emails, or spam, written to appear as if they have been sent by banks or other reputable organizations

Somatosensory and supra-spinal endogenous inhibitory functions were assessed in stroke patients with persistent PSSP (n=19), pain-free stroke patients (PF, n=29) and healthy

Het werkvak is niet altijd goed afgesloten, het is niet altijd duidelijk welke gedragsaanpassingen van fietsers verwacht worden en het bord 'fietser afstappen' wordt soms

The present text seems strongly to indicate the territorial restoration of the nation (cf. It will be greatly enlarged and permanently settled. However, we must

IMPLICATIONS FOR CUSTOMER LIFETIME VALUE AND CUSTOMER EQUITY To examine how the estimated effects influence CLV, we compare the CLV of free-trial and regular customers, and

Using this profile, we sought to model the three most pro- nounced narrow features for PbS_BDT between 0.12 and 0.19 eV (consider the Supporting Information for a detailed