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A queueing network model for field service support systems
Omega, Int. J. Mgmt Sci. Vol.22, No. I, pp. 35-40, 1994 Copyright © 1994Elsevier Science Ltd Printed in Great Britain.All rightsreserved
Pergamon
0305-0483/94$6.00+ 0.00
A Queueing Network Model for Field Service Support Systems AAW
WALLER
Industrial Engineering, Auburn University, Minneapolis, USA (Received 17 August 1993; in revised form 27 September 1993) This paper combines research in the area of computer systems performance modeling with the area of fidd service support systems. Field service support is becoming increasingly important with the rising complexity of the products in many industries. In particular, the field service of communication equipment, manufacturing equipment and computers is a major factor of consumer choice. Managers of field service support systems need models and tools to analyze the impact of strategic decisions on customer service and cost. This paper presents a useful, computationally effglent model to evaluate the impact of many strategic decisions, it provides a strong example of the integrative research which is vital to industries and the practice of management science. Important extensions and other areas of related research are highlighted as well.
Key words~ueueing, strategic planning, field service, steady-state analysis
1. INTRODUCTION
parts inventory [1-3, 10, 13]. Others consider the spare parts inventory planning, while not including the staffing issue [7, 10, 11, 13]. One author considers both issues, but that model is not computationally efficient [14]. In developing the model below, our objective is to create a simple, tractable model which incorporates staffing, spare part availability and an emergency delivery option. In addition, we wish to examine the influence of increased investment in inventory on customer service levels. Finally, we desire a model which could easily include more detailed sub-models to reflect the peculiarities of particular problem instances.
PROVIDING TIMELY FIELD SERVICE has become a priority for many industries, such as the communication, computer and photocopier industries. The increasing sophistication of the products in these industries has resulted in higher demand for original manufacturer maintenance and repair service and in less reliance on an internal 'handyman' to repair the products. The management of a field service support system (FSSS) to provide high quality, low cost service is a complex task. The three primary decisions are how many field engineers (FEs) are required, how should customers be allocated to the FEs, and how the spare parts inventory should be managed. In this paper, we present a closed queueing network model which enables managers to examine the impact on customer service of various policies and decisions related to these three decisions. This model is easy to implement and is computationally efficient. Several authors examine FSSS from different perspectives. Some authors address the question of staff size, while not considering the spare O~E 22/I--D
2. MODEL DEVELOPMENT In this paper, we will concentrate on FSSS with the following characteristics. A single FE is responsible for a given set of customers in order to enhance the manufacturer-customer relationship and to protect customers' confidentiality. The spare parts kit carried by the FE results in 35
WaUer--A QueueingNetwork Model
36
a given percentage of the repairs being completed with the kit stock. If a part is not available in the kit, it is ordered from a depot and delivered directly to the customer. Such a delivery will be referred to as an 'emergency delivery.' In this case, the repair cannot be completed during the first visit, so the repair is considered 'broken' and the FE leaves to service another customer. Customers are serviced in a first-in-first-out manner. In addition, we assume that the service time for each customer, which includes travel time and on-site time, is exponentially distributed. However, the emergency delivery time and the time between equipment failures are allowed to be from any distribution with a rational Laplace transform. The modeling approach is easily extented to allow heterogeneous customers, each with different service means; but, for notational convenience, we will assume a common mean. The FSSS is modeled as a Baskett, Chandy, Muntz and Palacois (BCMP) queueing network [5, 6] with two classes of customers. One class reflects those customers waiting for an initial visit from the FE and the other represents those customers whose emergency delivered parts have arrived and they are waiting for the FE to return to install the parts. The model is illustrated in Fig. 1. A customer cycles through the system, beginning at node 0,
with the equipment running within specifications ('up' customers). When a breakdown occurs, the customer enters node 1 as a class 1 customer, waiting for the FE's initial visit. The customer is eventually serviced by the FE. With probability PE, the FE does not have the parts necessary to complete the repair. Therefore, the FE would place an emergency order for the part and leave to service another customer. The original customer would enter node 2 to await delivery of the part. When the part arrives, the customer notifies the FE and enters node 1 as a class 2 (return visit) customer. When the FE returns, the repair is completed and the customer returns to node 0. In the event that the FE does have the spare parts on the first visit, the customer by-passes the emergency delivery and returns directly to node 0 as an up customer. In the Appendix, we analyze the queueing network model and develop closed form solutions for the expected number of customers at each node and for the expected idle time of the FE. 3. APPLICATION OF THE MODEL FSSS managers have three primary opportunities to invest capital in the system to improve the service: increase the spare parts inventory, hire more FEs and/or decrease the emergency delivery time. In this section, we will explore Node 2 Emergency Delivery Customers
~]
/Node 0 "Up" Customers
M/M/
M/M/
ll,I
®
<--\
ReturnVisit~ I \ ,, Node 1 | \ Waiting/In Repair" ]
Customers M/ M/ I
/PE Fi~t Visit/ -
oo
FirstVisit
~
Ill ] I.......I.--.i,I--pE "1m ReturnVisit/
Fig. 1. BCMP network for FSSS.
Omega, Vol. 22, No. 1
these opportunities using a hypothetical FSSS, derived from a real company. Explicit detail about the company is unavailable owing to confidentiality concerns. For this hypothetical system, the emergency delivery time is 8 h, the mean service time for an initial visit is 3 h, the mean service time for a return visit is 1 h, the mean time between failures for a particular customer is 200 h, and the probability of not having the part in the spare parts kit is 0.3. In addition, there are four FEs with 50 customers each. In the model, the inventory investment is represented by the parameter PE, the probability that the FE will not have the necessary parts, so that greater investment in inventory will reduce PE. Figure 2 illustrates graphically the effect of PE on the percentage of customers which are 'up' for various levels of the emergency delivery time. As expected, PE has a larger effect when the emergency delivery time is longer. Suppose we wanted to increase the percentage of 'up' customers for the current level of 93.3% to at least 95°/0. One option is to invest in enough spare parts inventory to decrease PE from 0.3 to 0.05. A second option is to achieve this increase in customer service by decreasing PE to 0.10 and simultaneously decreasing the emergency delivery time to 2 h. The FSSS manager can use graphs such as these to generate various options to improve the system. Choices among the options can then be made on intangible criteria. Figure 3 shows the effect of PE on the probability that the FE is idle. The plot shows that our first option above would result in an increase in the FE idle time of approx. 4%. Our second option would result in an increase of 3°/0 idle time. Therefore, either option would 96
-
O
o
92 90 88
o TE = 2 A TE = 8 * TE = 16
86
"*~. ~',~ ~ ,
"~
I
I
I
I
I
T
0
0.2
0.4
0.6
0.8
1.0
Probability of e m e r g e n c y delivery Fig, 2. Percentage of 'up" customers.
37
25 to t~
20
15
o TE=2 ATE= 8 *TE= 16 I 0.2
~ ~ , ~ , ~N~x'~ I 0.4
I 0.6
I 0.8
~'' Y 1.0
Probability of e m e r g e n c y delivery Fig. 3. FE idle time percentage.
be reasonable with respect to the load on the FEs. Figure 3 also indicates that the FE idle time is more sensitive to PE when the emergency delivery time is smaller. In order to use the model to assess the impact of a specific dollar value of inventory investment, the manager needs a more detailed submodel which translates inventory investment dollars and a specific inventory management policy into the job completion rate. The FSSS manager may select the spare parts kitting model which best fits the tactical operating policies of the system, including restocking policies and service discipline policies. In addition to increasing the inventory investment, the FSSS manager may also consider hiring more FEs. Field service territory design is a complicated issue, especially when there are various types of customers requiring different levels of service. One goal of territory design is to assign to each FE a collection of customers large enough to utilize the FE effectively, but small enough to maintain a high level of service. With the model described above, we can calculate the maximum number of customers which can be assigned to an FE while maintaining a specified high service level. The formula for computing this maximum number of customers is given in the Appendix. Note that the number of customers assigned to an FE will affect the mean service time through the increase in travel time between customers. The routing, dispatching and scheduling literature provides models for capturing this increase, for example, Berman et al. [5]. In Fig. 4, we plot the maximum number of customers versus PE for our hypothetical system with a service level of 95%. For each emergency
38
Waller--A Queueing Network Model I
40
N=s0
\
o 0.2
0.4
0.6
0.8
1.0
Probability of emergency delivery Fig. 4. Maximum number of customers--95% service level.
delivery time, the maximum number of customers is quite sensitive to PE and the emergency delivery time. The maximum number of customers can be used to calculate a lower bound on the number of FEs which are required to maintain a given service level. For both options above, 4 FEs are sufficient to maintain a 95% service level, since the maximum number of customers is 52 and 51, respectively. However, we could also consider leaving the emergency delivery time and PE at their current levels and increasing the number o f FEs to 5, resulting in a maximum of 41 customers per FE. We have demonstrated how these different options for increasing the customer service level to 95% can be analyzed graphically with respect to customer service and FE idle time. The third opportunity for investing capital into the FSSS is to reduce the emergency delivery time. This reduction can be accomplished in many ways. For example, a same-day courier service could be employed for the emergency deliveries. A more complex solution is to establish local depots which service several FE terri-
96 o
94 92
)
90 delivery time = 8
N 88 86 84
-t~
-*~.
~xN = 45 *N=50 oN=55 0.2
-~
~* "~
I
I
1
0.4
0.6
0.8
"~ 1.0
Probability of emergency delivery Fig. 5. Percentage of 'up' customers--varying N.
tories. In this case, the question of how to stock the depot and its effect on PE and the emergency delivery time becomes crucial. Each of the Figs 2-4 show that reducing the emergency delivery time improves customer service and decreases the FE's idle time. In addition to aiding strategic decisions over which the FSSS manager typically has control, the model can also aid in the strategic management of other decisions. For example, suppose the manufacturing company has decided to introduce a new line of products which will increase the number of customers being served. The FSSS manager needs to be able to evaluate the potential impact of this new product line on overall customer service. Figure 5 shows the percentage of operational customers for different levels of N, the number of customers assigned to each FE. The horizontal line indicates the 95% customer service level. For a constant PE, Fig. 5 illustrates that the customer service declines as the number of customers increases. For example, at the current values of 0.3, when the number of customers per FE increases from 50 to 55, the percentage of 'up' customers decreases from 93.3 to 91.9%. Figure 5 also enumerates 14 different combinations of PE and N which achieve the target 95% level. The FSSS manager can then examine these satisficing combinations and evaluate which of them are more desirable in terms of criteria which are difficult to quantify. Plots analogous to Fig. 5 can provide justification for increasing inventory investment to reduce PE or for a larger FE staff size to reduce the number of customers per FE. 4. CONCLUSIONS AND FURTHER RESEARCH We have addressed strategic planning issues in field service support systems by developing a closed queueing network model. We have derived analytic formulae for steady-state performance measures and shown how different parameters affect these measures. The model is easly to implement and Figs 2-5 illustrate graphical methods that show how the model can be a powerful tool for strategic allocation of resources by a FSSS manager to improve customer service. The model opens the door to further work on FSSS. Some important areas of research include: exploring the use of various sub-models
Omega, Vol. 22, No. 1
to estimate PE as a function of the spare parts inventory policies; investigation of the expansion of the model to include a local depot which serves several FEs and carries a limited inventory; and evaluating various approximations to other measures of service, such as the expected time that a customer is down and the expected time that a customer waits for the initial visit of the FE. There is additional opportunity for research in the development of a model to estimate service times for the initial and return visits under different inventory structures, perhaps incorporating models from routing and scheduling. Other avenues for future research include the modeling of multiple FE territories with state-dependent routing; and the development of a total system cost model.
39
Step 2: Calculate the product function for each node. (w0)~0 f0(n0) = - - , f~(hi) = (w,)~', .A(n2)= (w'~)~" n0! n2!
Step 3" Calculate the normalization constant. N
N-hi
G= ~, ~ nl
fofno)A(ni)A(N-no-n,)
= 0 #10=0
jSot
Step 4: Calculate the marginal state distribution of the joint number of customers at the nodes. n (n0, hi, N -- n o - n I ) = G - If0(n 0)fl (nl)f2 (N - n o - rt I ).
Using the marginal state distribution, we can calculate the expected number of customers at each node.
APPENDIX
k=O~WlJk !
We now show how the model is analyzed as a closed queueing network. First we define notation used in the analysis. N = the total number of customers in the system no = the number of 'up' customers nl = the number of customers waiting for or in repair n2 = the number of customers waiting for an emergency delivery T~ = the mean time between failures for a customer TE = the mean emergency delivery time Tr~ = the mean service (travel, diagnosis, and repair) time for an initial visit TR2 = the mean service (travel and replacement) time for a return visit PE = the probability that the FE does not have the necessary spare parts to complete a repair during the initial visit. The analysis of the network model depicted in Fig. 1 is straightforward. The network is a B C M P network [4]. The steps for analyzing a B C M P network in general are detailed in Conway and Georganas [6]. We present the applications of these steps to our model.
Step 1: Calculate the relative traffic intensity for each node. w°
r~ 2(I +PE)'
Wl
TR, + p~ TE 2(I +Pe)
):
and
w2
peTE 2(I +Pe)"
N-
I
E(n,)=G-'(w,)" T_. (n-k
j=o
\w,Jj!;
.fW 0 + V"2"~k I
Im.
:
and
k=O
Iw\N-I/w\k
1 N-I-klw\]I
We can also calculate the probability that the FE is idle: 1 P {n I = 0} = G-I ~ (w° + wz)N.
Finally, the maximum number of customers that may be assigned to an FE, while ensuring that, on average, at least • percent of the assigned customers are operational, is the largest N that satisfies the following equation: IW\N-tIW
\k l
e(.0)=c-'(w,)"/"°/Z \wu k=0 \ w d
X
K, n - t - k / w 2\J l Z l1 j=o \ 'i/ 3.
>~
ctN
.
100
REFERENCES 1. Agnihothri SR (1985) Performance evaluation of field service territories. Unpublished Ph.D. dissertation, Graduate School of Management, The University of Rochester. 2. Agnihothri SR (1988) A mean value analysis of the traveling repairman problem, l i E Trans. 20(2), 223-229. 3. Agnihothri SR (1989) Interrelationships between performance measures for the machine-repairman problem. Nay. Res. Logist. 36, 265-271. 4. Baskett F, Chandy K M , Muntz R R and Palacios F (1975) Open, closed and mixed networks of queues with different classes of customers. J. Ass. Comput. Machin. 22, 248-260.
40
Waller--A Queueing Network Model
5. Berman O, Larson RC and Chiu SS (1985) Optimal server location on a network operating as an M/G/I queue. Ops Res. 33(4), 746-77t. 6. Conway AE and Georganas ND 0989) Queueing Networks--Exact Computational Algorithms: A Unified Theory Based on Decomposition and Aggregation. The MIT Press, Cambridge, Mass. 7. Graves SC (1982) A multiple-item inventory model with a job completion criterion. Mgmt Sci. 28(11), 1334-1337. 8. Graves SC (1988) Determining the spares and staffing levels for a repair depot. J. Manufact. Ops Mgmt 1(2), 227-24 I. 9. Hambleton RS (1982) A manpower planning model for mobile repairmen. J. Opl Res. Soc. 33, 621-627. 10. Mamer JW and Smith SA (1982) Optimizing field repair kits based on job completion rate. Mgmt Sci. 28(11), 1328-1333.
II. Mamer JW and Smith SA (1985) Job completion based inventory systems: optimal policies for repair kits and spare machines. Mgmt Sci. 31(6), 703-718. 12. Smith SA (1979) Estimating service territory size. Mgmt Sci. 25(4), 301-311. 13. Smith SA, Chambers JC and Shilfer E (1980) Optimal inventories based on job completion rate for repairs requiring multiple items. Mgmt Sci. 26(8), 849-852. 14. Waller AAW (1991) Strategic planning for field service support systems. Unpublished Ph.D. dissertation, Cornell University. Professor AA W Waller, 5028 Nicollet Avenue South, Minneapolis, MN 55419, USA.
ADDRESS FOR CORRESPONDENCE:
Pergamon
0305-0483/94$6.00+ 0.00
A Queueing Network Model for Field Service Support Systems AAW
WALLER
Industrial Engineering, Auburn University, Minneapolis, USA (Received 17 August 1993; in revised form 27 September 1993) This paper combines research in the area of computer systems performance modeling with the area of fidd service support systems. Field service support is becoming increasingly important with the rising complexity of the products in many industries. In particular, the field service of communication equipment, manufacturing equipment and computers is a major factor of consumer choice. Managers of field service support systems need models and tools to analyze the impact of strategic decisions on customer service and cost. This paper presents a useful, computationally effglent model to evaluate the impact of many strategic decisions, it provides a strong example of the integrative research which is vital to industries and the practice of management science. Important extensions and other areas of related research are highlighted as well.
Key words~ueueing, strategic planning, field service, steady-state analysis
1. INTRODUCTION
parts inventory [1-3, 10, 13]. Others consider the spare parts inventory planning, while not including the staffing issue [7, 10, 11, 13]. One author considers both issues, but that model is not computationally efficient [14]. In developing the model below, our objective is to create a simple, tractable model which incorporates staffing, spare part availability and an emergency delivery option. In addition, we wish to examine the influence of increased investment in inventory on customer service levels. Finally, we desire a model which could easily include more detailed sub-models to reflect the peculiarities of particular problem instances.
PROVIDING TIMELY FIELD SERVICE has become a priority for many industries, such as the communication, computer and photocopier industries. The increasing sophistication of the products in these industries has resulted in higher demand for original manufacturer maintenance and repair service and in less reliance on an internal 'handyman' to repair the products. The management of a field service support system (FSSS) to provide high quality, low cost service is a complex task. The three primary decisions are how many field engineers (FEs) are required, how should customers be allocated to the FEs, and how the spare parts inventory should be managed. In this paper, we present a closed queueing network model which enables managers to examine the impact on customer service of various policies and decisions related to these three decisions. This model is easy to implement and is computationally efficient. Several authors examine FSSS from different perspectives. Some authors address the question of staff size, while not considering the spare O~E 22/I--D
2. MODEL DEVELOPMENT In this paper, we will concentrate on FSSS with the following characteristics. A single FE is responsible for a given set of customers in order to enhance the manufacturer-customer relationship and to protect customers' confidentiality. The spare parts kit carried by the FE results in 35
WaUer--A QueueingNetwork Model
36
a given percentage of the repairs being completed with the kit stock. If a part is not available in the kit, it is ordered from a depot and delivered directly to the customer. Such a delivery will be referred to as an 'emergency delivery.' In this case, the repair cannot be completed during the first visit, so the repair is considered 'broken' and the FE leaves to service another customer. Customers are serviced in a first-in-first-out manner. In addition, we assume that the service time for each customer, which includes travel time and on-site time, is exponentially distributed. However, the emergency delivery time and the time between equipment failures are allowed to be from any distribution with a rational Laplace transform. The modeling approach is easily extented to allow heterogeneous customers, each with different service means; but, for notational convenience, we will assume a common mean. The FSSS is modeled as a Baskett, Chandy, Muntz and Palacois (BCMP) queueing network [5, 6] with two classes of customers. One class reflects those customers waiting for an initial visit from the FE and the other represents those customers whose emergency delivered parts have arrived and they are waiting for the FE to return to install the parts. The model is illustrated in Fig. 1. A customer cycles through the system, beginning at node 0,
with the equipment running within specifications ('up' customers). When a breakdown occurs, the customer enters node 1 as a class 1 customer, waiting for the FE's initial visit. The customer is eventually serviced by the FE. With probability PE, the FE does not have the parts necessary to complete the repair. Therefore, the FE would place an emergency order for the part and leave to service another customer. The original customer would enter node 2 to await delivery of the part. When the part arrives, the customer notifies the FE and enters node 1 as a class 2 (return visit) customer. When the FE returns, the repair is completed and the customer returns to node 0. In the event that the FE does have the spare parts on the first visit, the customer by-passes the emergency delivery and returns directly to node 0 as an up customer. In the Appendix, we analyze the queueing network model and develop closed form solutions for the expected number of customers at each node and for the expected idle time of the FE. 3. APPLICATION OF THE MODEL FSSS managers have three primary opportunities to invest capital in the system to improve the service: increase the spare parts inventory, hire more FEs and/or decrease the emergency delivery time. In this section, we will explore Node 2 Emergency Delivery Customers
~]
/Node 0 "Up" Customers
M/M/
M/M/
ll,I
®
<--\
ReturnVisit~ I \ ,, Node 1 | \ Waiting/In Repair" ]
Customers M/ M/ I
/PE Fi~t Visit/ -
oo
FirstVisit
~
Ill ] I.......I.--.i,I--pE "1m ReturnVisit/
Fig. 1. BCMP network for FSSS.
Omega, Vol. 22, No. 1
these opportunities using a hypothetical FSSS, derived from a real company. Explicit detail about the company is unavailable owing to confidentiality concerns. For this hypothetical system, the emergency delivery time is 8 h, the mean service time for an initial visit is 3 h, the mean service time for a return visit is 1 h, the mean time between failures for a particular customer is 200 h, and the probability of not having the part in the spare parts kit is 0.3. In addition, there are four FEs with 50 customers each. In the model, the inventory investment is represented by the parameter PE, the probability that the FE will not have the necessary parts, so that greater investment in inventory will reduce PE. Figure 2 illustrates graphically the effect of PE on the percentage of customers which are 'up' for various levels of the emergency delivery time. As expected, PE has a larger effect when the emergency delivery time is longer. Suppose we wanted to increase the percentage of 'up' customers for the current level of 93.3% to at least 95°/0. One option is to invest in enough spare parts inventory to decrease PE from 0.3 to 0.05. A second option is to achieve this increase in customer service by decreasing PE to 0.10 and simultaneously decreasing the emergency delivery time to 2 h. The FSSS manager can use graphs such as these to generate various options to improve the system. Choices among the options can then be made on intangible criteria. Figure 3 shows the effect of PE on the probability that the FE is idle. The plot shows that our first option above would result in an increase in the FE idle time of approx. 4%. Our second option would result in an increase of 3°/0 idle time. Therefore, either option would 96
-
O
o
92 90 88
o TE = 2 A TE = 8 * TE = 16
86
"*~. ~',~ ~ ,
"~
I
I
I
I
I
T
0
0.2
0.4
0.6
0.8
1.0
Probability of e m e r g e n c y delivery Fig, 2. Percentage of 'up" customers.
37
25 to t~
20
15
o TE=2 ATE= 8 *TE= 16 I 0.2
~ ~ , ~ , ~N~x'~ I 0.4
I 0.6
I 0.8
~'' Y 1.0
Probability of e m e r g e n c y delivery Fig. 3. FE idle time percentage.
be reasonable with respect to the load on the FEs. Figure 3 also indicates that the FE idle time is more sensitive to PE when the emergency delivery time is smaller. In order to use the model to assess the impact of a specific dollar value of inventory investment, the manager needs a more detailed submodel which translates inventory investment dollars and a specific inventory management policy into the job completion rate. The FSSS manager may select the spare parts kitting model which best fits the tactical operating policies of the system, including restocking policies and service discipline policies. In addition to increasing the inventory investment, the FSSS manager may also consider hiring more FEs. Field service territory design is a complicated issue, especially when there are various types of customers requiring different levels of service. One goal of territory design is to assign to each FE a collection of customers large enough to utilize the FE effectively, but small enough to maintain a high level of service. With the model described above, we can calculate the maximum number of customers which can be assigned to an FE while maintaining a specified high service level. The formula for computing this maximum number of customers is given in the Appendix. Note that the number of customers assigned to an FE will affect the mean service time through the increase in travel time between customers. The routing, dispatching and scheduling literature provides models for capturing this increase, for example, Berman et al. [5]. In Fig. 4, we plot the maximum number of customers versus PE for our hypothetical system with a service level of 95%. For each emergency
38
Waller--A Queueing Network Model I
40
N=s0
\
o 0.2
0.4
0.6
0.8
1.0
Probability of emergency delivery Fig. 4. Maximum number of customers--95% service level.
delivery time, the maximum number of customers is quite sensitive to PE and the emergency delivery time. The maximum number of customers can be used to calculate a lower bound on the number of FEs which are required to maintain a given service level. For both options above, 4 FEs are sufficient to maintain a 95% service level, since the maximum number of customers is 52 and 51, respectively. However, we could also consider leaving the emergency delivery time and PE at their current levels and increasing the number o f FEs to 5, resulting in a maximum of 41 customers per FE. We have demonstrated how these different options for increasing the customer service level to 95% can be analyzed graphically with respect to customer service and FE idle time. The third opportunity for investing capital into the FSSS is to reduce the emergency delivery time. This reduction can be accomplished in many ways. For example, a same-day courier service could be employed for the emergency deliveries. A more complex solution is to establish local depots which service several FE terri-
96 o
94 92
)
90 delivery time = 8
N 88 86 84
-t~
-*~.
~xN = 45 *N=50 oN=55 0.2
-~
~* "~
I
I
1
0.4
0.6
0.8
"~ 1.0
Probability of emergency delivery Fig. 5. Percentage of 'up' customers--varying N.
tories. In this case, the question of how to stock the depot and its effect on PE and the emergency delivery time becomes crucial. Each of the Figs 2-4 show that reducing the emergency delivery time improves customer service and decreases the FE's idle time. In addition to aiding strategic decisions over which the FSSS manager typically has control, the model can also aid in the strategic management of other decisions. For example, suppose the manufacturing company has decided to introduce a new line of products which will increase the number of customers being served. The FSSS manager needs to be able to evaluate the potential impact of this new product line on overall customer service. Figure 5 shows the percentage of operational customers for different levels of N, the number of customers assigned to each FE. The horizontal line indicates the 95% customer service level. For a constant PE, Fig. 5 illustrates that the customer service declines as the number of customers increases. For example, at the current values of 0.3, when the number of customers per FE increases from 50 to 55, the percentage of 'up' customers decreases from 93.3 to 91.9%. Figure 5 also enumerates 14 different combinations of PE and N which achieve the target 95% level. The FSSS manager can then examine these satisficing combinations and evaluate which of them are more desirable in terms of criteria which are difficult to quantify. Plots analogous to Fig. 5 can provide justification for increasing inventory investment to reduce PE or for a larger FE staff size to reduce the number of customers per FE. 4. CONCLUSIONS AND FURTHER RESEARCH We have addressed strategic planning issues in field service support systems by developing a closed queueing network model. We have derived analytic formulae for steady-state performance measures and shown how different parameters affect these measures. The model is easly to implement and Figs 2-5 illustrate graphical methods that show how the model can be a powerful tool for strategic allocation of resources by a FSSS manager to improve customer service. The model opens the door to further work on FSSS. Some important areas of research include: exploring the use of various sub-models
Omega, Vol. 22, No. 1
to estimate PE as a function of the spare parts inventory policies; investigation of the expansion of the model to include a local depot which serves several FEs and carries a limited inventory; and evaluating various approximations to other measures of service, such as the expected time that a customer is down and the expected time that a customer waits for the initial visit of the FE. There is additional opportunity for research in the development of a model to estimate service times for the initial and return visits under different inventory structures, perhaps incorporating models from routing and scheduling. Other avenues for future research include the modeling of multiple FE territories with state-dependent routing; and the development of a total system cost model.
39
Step 2: Calculate the product function for each node. (w0)~0 f0(n0) = - - , f~(hi) = (w,)~', .A(n2)= (w'~)~" n0! n2!
Step 3" Calculate the normalization constant. N
N-hi
G= ~, ~ nl
fofno)A(ni)A(N-no-n,)
= 0 #10=0
jSot
Step 4: Calculate the marginal state distribution of the joint number of customers at the nodes. n (n0, hi, N -- n o - n I ) = G - If0(n 0)fl (nl)f2 (N - n o - rt I ).
Using the marginal state distribution, we can calculate the expected number of customers at each node.
APPENDIX
k=O~WlJk !
We now show how the model is analyzed as a closed queueing network. First we define notation used in the analysis. N = the total number of customers in the system no = the number of 'up' customers nl = the number of customers waiting for or in repair n2 = the number of customers waiting for an emergency delivery T~ = the mean time between failures for a customer TE = the mean emergency delivery time Tr~ = the mean service (travel, diagnosis, and repair) time for an initial visit TR2 = the mean service (travel and replacement) time for a return visit PE = the probability that the FE does not have the necessary spare parts to complete a repair during the initial visit. The analysis of the network model depicted in Fig. 1 is straightforward. The network is a B C M P network [4]. The steps for analyzing a B C M P network in general are detailed in Conway and Georganas [6]. We present the applications of these steps to our model.
Step 1: Calculate the relative traffic intensity for each node. w°
r~ 2(I +PE)'
Wl
TR, + p~ TE 2(I +Pe)
):
and
w2
peTE 2(I +Pe)"
N-
I
E(n,)=G-'(w,)" T_. (n-k
j=o
\w,Jj!;
.fW 0 + V"2"~k I
Im.
:
and
k=O
Iw\N-I/w\k
1 N-I-klw\]I
We can also calculate the probability that the FE is idle: 1 P {n I = 0} = G-I ~ (w° + wz)N.
Finally, the maximum number of customers that may be assigned to an FE, while ensuring that, on average, at least • percent of the assigned customers are operational, is the largest N that satisfies the following equation: IW\N-tIW
\k l
e(.0)=c-'(w,)"/"°/Z \wu k=0 \ w d
X
K, n - t - k / w 2\J l Z l1 j=o \ 'i/ 3.
>~
ctN
.
100
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40
Waller--A Queueing Network Model
5. Berman O, Larson RC and Chiu SS (1985) Optimal server location on a network operating as an M/G/I queue. Ops Res. 33(4), 746-77t. 6. Conway AE and Georganas ND 0989) Queueing Networks--Exact Computational Algorithms: A Unified Theory Based on Decomposition and Aggregation. The MIT Press, Cambridge, Mass. 7. Graves SC (1982) A multiple-item inventory model with a job completion criterion. Mgmt Sci. 28(11), 1334-1337. 8. Graves SC (1988) Determining the spares and staffing levels for a repair depot. J. Manufact. Ops Mgmt 1(2), 227-24 I. 9. Hambleton RS (1982) A manpower planning model for mobile repairmen. J. Opl Res. Soc. 33, 621-627. 10. Mamer JW and Smith SA (1982) Optimizing field repair kits based on job completion rate. Mgmt Sci. 28(11), 1328-1333.
II. Mamer JW and Smith SA (1985) Job completion based inventory systems: optimal policies for repair kits and spare machines. Mgmt Sci. 31(6), 703-718. 12. Smith SA (1979) Estimating service territory size. Mgmt Sci. 25(4), 301-311. 13. Smith SA, Chambers JC and Shilfer E (1980) Optimal inventories based on job completion rate for repairs requiring multiple items. Mgmt Sci. 26(8), 849-852. 14. Waller AAW (1991) Strategic planning for field service support systems. Unpublished Ph.D. dissertation, Cornell University. Professor AA W Waller, 5028 Nicollet Avenue South, Minneapolis, MN 55419, USA.
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