Fork-Join Networks in Heavy Traffic:

Diffusion Approximations and Control

M.Sc. Research Proposal

Asaf Zviran

Advisors: Prof. Rami Atar

Prof. Avishai Mandelbaum

Faculty of Industrial Engineering and Management

Technion—Israel Institute of Technology

August 20, 2008

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Contents

1 Introduction 3

1.1 Fork-Join Networks: Definition and Some Applications . . . . . . . . . . . 3

1.2 Short Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Preliminaries and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Mathematical Model 8

2.1 System Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 State Space Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Fluid and Diffusion Limits 17

3.1 Fluid Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Diffusion Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 State-Space Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Heavy Traffic Limit For The Throughput Time . . . . . . . . . . . . . . . 24

4 Brief Discussion on System Control 27

4.1 Length of Stay Estimation—The Snapshot Principle . . . . . . . . . . . . . 27

4.2 Priorities Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Staffing-Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Proposed Research 33

5.1 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Stochastic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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1 Introduction

This research proposal deals with the development of heavy traffic approximations, which

enables some practical aspects of analysis and control of system behavior in fork-join

networks. Despite the importance of parallel processing of the kind which appears in the

fork-join class, there are still few results concerning many practical and important aspects

of the system’s behavior. In Section 1.1 we give the definition and some examples of fork

join networks and their application. In Section 1.2 we give a short survey of the work

that has been done in developing methods to analyze that family of networks. In Section

1.3 we start with notations and definitions in preparation for the system analysis in the

following sections.

1.1 Fork-Join Networks: Definition and Some Applications

A fork-join network consists of a group of service stations, which serve the arriving cus-

tomers simultaneously and sequently according to preset deterministic precedence con-

straints. More specifically, one can think in terms of “jobs” arriving to the system over

time, each job containing different tasks that need to be executed according to the prece-

dence constraints. The job may leave the system only after all its tasks have finished

their service. The distinguishing features of this model class are the so-called “fork” and

“join” constructs. A fork occurs whenever several tasks are being processed at the same

time. In the network model, this is represented by a “splitting” of the job into multiple

tasks, which are then sent simultaneously to their respective servers. A join node, on the

other hand, corresponds to a task that may not be initiated until several other tasks have

been completed. Components are joined only if they correspond to the same job; thus a

join is always preceded by a fork. If the last stage of operation consists of multiple tasks,

then these tasks regroup into a single job before departing the system.

Fork-Join task flow examples

In Fig. 1.1 we can see the process progressing from the Arrest of an “alleged” criminal until

getting him to trial (arraignment). As shown, the process consists of three simultaneous

paths—the path of the arrestee, the path of the arresting officer, and the path of the

arrestee information through the system. This example is taken from Larson’s article [7]

on “improving the N.Y.C A-to-A system”.

In Fig. 1.2 we can see the process taking place from the arrival of an order to build a

house until the completion of the numerous tasks required in the construction plan. In

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the graph, the construction order “split” and “join” throughout the system till all the

tasks are finished; the precedence constraints take the form of a flow chart.

Figure 1.1: Arrest-to-Arraignment Process

Figure 1.2: Construction of a house

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Fork-Join networks are natural models for a variety of processes including communica-

tion and computer systems, manufacturing and project management (as introduced in Fig.

1.2) and service systems (as introduced in Fig. 1.1). A fork-join computer or telecommu-

nication network typically represents the processing of computer programs, data packets,

etc., which involve parallel multitasking and the splitting and joining of information. In

manufacturing, a fork-join network, called an assembly network, represents the assembly

of a product or system which requires several parts that are processed simultaneously at

separate workstations or plant locations.

Fork-join networks can be found frequently in the health-care system in general, and hos-

pitals in particular (see Fig. 1.3), in which the patient and his medical file, test results,

insurance policy may split and join in different parts of the process in order to get to

the final task, that may be admitting a patient to the wards, starting an operation, etc.

Another reason for the need of a fork-join network in hospitals is the necessity to join and

synchronize many separate resources—doctors, nurses, room/bed, special equipment—in

order to perform one integrated operation. In this research we will try to develop and

implement the mathematical approximations needed for some practical aspects of analysis

and control of hospital processes.

Figure 1.3: Fork-join network in hospitals—preparation to surgery

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1.2 Short Literature Survey

This research focuses on the use of heavy traffic limits and especially fluid and diffusion

limits in the analysis of stochastic networks. In his book [5] Harrison presents the basics

for this work including the regulator mapping for stochastic processes, the representation

of the processes as Reflected Brownian Motion (RBM) and examples of optimal control in

Brownian Motion. The processes examined in his book were simple buffer flow processes.

His method for analyzing stochastic models by Brownian Motion was later expanded by

him in [6]. Chen and Yao present in their book [3] the method for constructing regulator

mapping for a G/G/1 system and developing the fluid and diffusion limits of queue-length

and workload processes. They show that the diffusion limit is one-dimensional RBM.

These two books ([5] and [3]) have had great impact on the development of the regulator

mapping and limit processes in Sections 2.1 and 3.

Nguyen developed in her papers [8] and [9] a system and state-space representation for

fork-join systems with homogeneous and heterogeneous customer population. She has

succeeded to show that those systems converge weakly in heavy traffic to multidimensional

RBM, whose state space is a nonsimple polyhedral cone within a nonnegative orthant. Her

system and state space representation are used in this proposal (Section 2.1) as the basis

for the limit processes development. In her work Nguyen assumed that all the stations in

the system converge to heavy traffic uniformly, which means that a state-space collapse

and a single bottleneck weren’t defined by her. Another assumption she used was that

the priority discipline is always FCFS. These assumptions are the baseline to the further

work that is being done in our research.

Peterson in his PhD thesis [10] worked on diffusion approximation for networks of queues

with multiple customer types. He focused on feedforward generalized Jackson systems

with preemptive discipline. He proved a heavy traffic limit theorem for these systems, with

multidimensional RBM. Peterson also didn’t include bottleneck analysis in his research

but he did show state-space collapse in the sense of customer type priorities. Reiman

and Simon [2] expanded Peterson’s work in the analysis of feedforward and feedbackward

generalized Jackson systems with multiple customer types. They included in their work an

analysis of single bottleneck systems, and showed that the systems converge weakly to one-

dimensional RBM. They also showed the effects of a state-space collapse of non-bottleneck

stations and a state-space collapse of high priority customers, and presented the notion

of the snapshot principle, which is used in Section 4.1. One of the goals in this proposal

is to expand their results to fork-join systems. During the work Whitt’s [11] paper was

helpful for the development of weak convergence of the processes of interest in diffusion

scaling. The paper from Banks and Dai [4] was used as a reference to the problems of

stability definition for systems which include feedback in their routing constraints.

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Cohen, Mandelbaum and Shtub [1] examined control mechanisms for project management

in a multi-project environment. They surveyed a variety of buffer management techniques

in open and closed systems. In this research we intend to examine the control problem

which arises in their work by means of optimal control constructed on the associated limit

process.

1.3 Preliminaries and Notations

The main subject of this proposal is the characterization of limiting processes for the

queue-length, total job-count and total workload processes in fork-join queues under heavy

traffic limits. Limit theorems that we use include the functional strong law of large

numbers (FSLLN) and the functional central limit Theorem (FCLT). Throughout this

proposal we shall follow the convention of using the terms fluid limit for the FSLLN limit

process and diffusion limit for the FCLT limit process.

Heavy traffic limits of the type considered in this proposal, which involve convergence of

normalized stochastic processes to a limit process (in this case multidimensional RBM),

utilize the notion of weak convergence of probability measures on metric spaces; the metric

space is Dr[0, 1], the r-dimension product space of right-continuous functions on [0, 1] that

have left limits, endowed with the Skorohod topology. Throughout the proposal weak

convergence is denoted by ⇒ distinguished from the almost surely convergence which is

denoted by →.

Throughout the proposal we will use the notation X(t) as the fluid limit of process X(t),

which satisfies the scaling Xn(t) = 1nX(nt) → X(t) u.o.c as n → ∞, and X(t) as the

diffusion limit which satisfies the scaling Xn(t) = n−1/2[Xn(t)− X(t)]⇒ X(t) as n→∞.

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2 Mathematical Model

In this section we develop the mathematical model for the processes of interest. Our

interest will be focused on a class of fork-join networks with single-server stations and

FCFS priority discipline; the customer flow into the system is homogeneous with a de-

terministic feedforward routing scheme. This system’s characterization has a feature for

preserving the ordering of customer arrivals, meaning the customer’s entrance order is

preserved throughout the system till the departure. This property has good implications

for the synchronization of tasks in join nodes.

We start in Section 2.1 with a pathwise construction of the job-count process vector and

total workload vector—the key performance measures of the system. In Section 2.2 we

develop the state space of the job-count process vector and prove that it is contained in

a nonsimple polyhedral cone.

2.1 System Representation

Characterization of The System Structure The network consists of J single-station

servers, indexed by j = 1, . . . , J. We assume that each server works at a constant rate,

and we denote by τj, the mean completion time of a job at station j. The network has

an input stream of homogeneous jobs and we denote by λ the average arrival rate of new

jobs.

The processing of each job requires the completion of J tasks. Each task is performed

at a specific single-server station, and the task performed at station j is referred to as

task j. The order in which tasks are performed is specified by a given set of precedence

constraints, which may allow some tasks to be performed in parallel and may require

that others be performed sequentially. Task i is said to be an immediate predecessor for

task j if upon completing task i the job moves immediately to station j. The precedence

relationships can be expressed via a precedence matrix P = (Pij) defined as follows:

Pij =

1, if task i is an immediate predecessor for task j,

0, otherwise.(2.1)

(Because all elements of the precedence matrix P are 0’s and 1’s, routing is clearly de-

terministic.) We assume that there is a column and row permutation of P such that the

resulting matrix is strictly upper triangular; in terms of the model, this means that we

consider only systems in which tasks are not repeated (feedforward systems).

Now we will define the buffers in the system. For every element Pij = 1 in the precedence

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matrix there is an associated buffer, and whenever a service is completed at station i,

one can think of the departing task as entering a waiting room or “buffer” preceding

server j. Notationally, it will be convenient to index the buffers by k = 1, . . . , K, when

every buffer k is associated with a different (i, j) pair. For each station j = 1, . . . , Jwe define β(j) as the set of buffers k that are incident to j. It can be seen that the

group of sets β(j) satisfies β(j) ≥ 1 ∀j and K ≡J∑j=1

β(j) ≥ J . The property β(j) > 1

characterizes servers j which are join nodes in the system, meaning the task that ar-

rives to k ∈ β(j) : β(j) > 1 may not be initiated until several other tasks performed

simultaneously in other servers merge with it in the “waiting rooms” of β(j). We let

s(k) ∈ 1 . . . J be the source of buffer k, that is, the station whose output feeds into

buffer k; if k is an external arrival buffer then we define s(k) ≡ 0. Accordingly we will

define s(β(j)) ⊂ 1 . . . J as the set of servers which are immediate predecessors to

station j; if j is an entrance server then we define s(β(j)) ≡ φ (empty set).

During the analysis of the system and the development of limit processes we will use

induction on the server stations from entrance stations till the departure stations. This

induction is based on the feedforward manner of the system, since in feedforward systems

we can argue that a station is not affected by the netflow of the stations following her

downstream but is affected by the netflow of her predecessor (upstream). The induction

is on d(j); the “depth” of station j which is defined in the following way. For j which his

immediate predecessors set is s(β(j)) ≡ φ, we set d(j) ≡ 1. Next, consider a station j

such that d(i) has been defined for all stations in its predecessor set s(β(j)). The depth

of station j is given by

d(j) ≡ maxd(i) : i ∈ s(β(j))+ 1 (2.2)

System Representation Example

Let’s define the system representation for the following system (see Fig. 2.1). The system

has 4 single server stations indexed by j ∈ 1, . . . , 4, and 7 buffers indexed by k ∈1, . . . , 7. The servers “depth” vector is represented as d = [1, 1, 2, 2]. The system

precedence matrix can be expressed as

P =

1 1 0 0 0

0 0 1 0 0

0 0 1 1 0

0 0 0 0 1

0 0 0 0 1

(2.3)

An example for the precedence sets may be: β(3) = 3, 4, and s(4) = 2.

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Figure 2.1: System Representation Example

System Primitive Data and Immediate Workload Processes The primitive data

that we use to construct the processes of interest are the i.i.d summation of interarrival

times and service times. Let (Ω,F , P ) be a probability space on which are defined se-

quences of random variables u(i), i ≥ 1 and vj(i), i ≥ 1, j = 1, . . . , J , where u(i)

and vj(i) are strictly positive with unit mean. For this proposal we will restrict ourselves

to the case where each is a sequence of i.i.d random variables and the J + 1 sequences

are mutually independent. From these sequences, the interarrival times and the service

times are constructed by setting the interarrival time of the i-th job to be λ−1u(i) and its

service time at station j to be τjvj(i).

We will assume that initially, i.e., at t = 0, the system is empty, meaning that Qk(0) =

0 ∀k. So the flow of jobs going through the system is characterized by two primitives:

External arrival process,

N(t) ≡ maxk :k∑i=0

λ−1u(i) ≤ t. (2.4)

Potential service process,

Sj(t) ≡ maxk :k∑i=0

τjvj(i) ≤ t. (2.5)

In addition, we assume (throughout the proposal) that the service discipline is work-

conserving, i.e., the stations cannot stay idle if there are complete jobs present at the

buffers preceding them.

Now in order to construct pathwise the immediate workload of the systems’ stations we

will use the induction on the stations depth (2.2). Starting with the entrance stations

(d(j) = 1) the station arrival process equals the external arrival process so the immediate

arrival process for j : d(j) = 1 is

Aj(t) ≡ N(t) ≡ maxk :k∑i=0

λ−1u(i) ≤ t. (2.6)

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Mj(t) - The immediate workload input process for station j : d(j) = 1, will be defined

as the sum of all service times at station j for customers or jobs that enter the network

during [0, t],

Mj(t) ≡ Vj(N(t)) =

N(t)∑i=0

τjvj(i), (2.7)

when Vj(N) is the partial sums process associated with the service times at the station,

Vj(N) ≡N∑i=0

τjvj(i).

Next we will set the immediate workload netflow process

ξj(t) = Mj(t)− t. (2.8)

because t is the potential amount of work that can be processed in t units of time1, ξj(t)

is the difference between the workload input and the potential workload output.

Let Qk(t) denote the number of jobs waiting to be processed in the buffer k ∈ β(j); we

shall refer to Qk(t) as the queue length process. Then for j : d(j) = 1 stations it should

be clear that |β(j)| ≡ 1 and the queue length process satisfy

Qk(t) = Ak(t)− Sj(Bj(t)) = Aj(t)− Sj(Bj(t)) (2.9)

Bj(t) =

∫ t

o

1 mink∈β(j)

Qk(s) > 0ds, (2.10)

where Bj(t) denotes the cumulative amount of time when the server is busy over the time

interval [0, t]; hence Sj(Bj(t)) is the number of jobs that have departed (after service

completion) from the system in the same time interval. We shall refer to Bj(t) as the

busy time process of server j, and define Dj(t) as the departure process

Dj(t) = Sj(Bj(t)) = maxk :k∑i=0

τjvj(i) ≤ Bj(t). (2.11)

Now we can define the immediate workload process

Wj(t) = Mj(t)−Bj(t), (2.12)

i.e., the sum of the impending service times for all jobs that are present in buffers incident

to j at time t, plus the remaining service time of any task that may be in service at time t.

In an inductive manner, these definitions can be extended to all stations in the net-

work. Consider a station j (d(j) > 1) such that all immediate predecessor stations have

1single-server stations

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been “treated”, i.e., their immediate processes have been defined. For such a station j

and for each buffer k ∈ β(j) one defines the immediate arrival process as the departure

process from its source Ak(t) = Ds(k)(t) ∀k ∈ β(j);

Aj(t) = mink∈β(j)

Ak(t).(2.13)

The rest of the process definitions can be extended in the same manner. In conclusion we

get an immediate process summary

Mj(t) =

Aj(t)∑i=0

τjvj(i).

ξj(t) = Mj(t)− t.

Dj(t) = Sj(Bj(t)) = maxk :k∑i=0

τjvj(i) ≤ Bj(t).

Qk(t) = Ak(t)− Sj(Bj(t)).

Wj(t) = Mj(t)−Bj(t).

(2.14)

That concludes the construction of the system’s immediate workload processes.

Multidimensional Regulator Mapping In this part we will apply a “centering”

operation on the immediate workload process of every station j separately, and rewrite it

as follows:Wj(t) = ξj(t) + Ij(t);

ξj(t) = Mj(t)− t;

Ij(t) = t−Bj(t) =∫ t

o1 min

k∈β(j)Qk(s) = 0ds;

(2.15)

When Ij(t) interprets as the cumulative amount of time the server is idle during [0, t], we

shall refer to Ij(t) as the idle time process.

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Furthermore, the following relation must hold: For all t ≥ 0 and all j,

Wj(t) ≥ 0;

dIj(t) ≥ 0, Ij(0) = 0;

Wj(t)dIj(t) = 0;

(2.16)

In other words, dIj(t) ≥ 0 means that Ij is nondecreasing; this is naturally satisfied, since

the idle time process Ij(t) is measured as cumulative over time. Wj(t)dIj(t) = 0 reflects

the work-conserving condition, the stations cannot be idle if there are complete jobs in

the stations’ queues waiting to be treated.

From now on we will denote W (t), ξ(t), I(t) as the immediate workload vector, immediate

netflow vector and the idle time vector respectively. The following theorem is stated for

multidimensional regulator mapping from ξ(t) to (W (t), I(t)).

Theorem 2.1 For any ξ(t) ∈ DJ , there exists a unique pair (W (t), I(t)) in D2×J simul-

taneously satisfying the following three properties for all j

Wj(t) = ξj(t) + Ij(t) ≥ 0;

dIj(t) ≥ 0, Ij(0) = 0;

Wj(t)dIj(t) = 0.

(2.17)

Furthermore, (Wj(t), Ij(t)) is given by

Ij(t) = Ψ(ξj(t)) = sup0≤s≤t

[−ξj(s)]+,

Wj(t) = Φ(ξj(t)) = ξj(t) + sup0≤s≤t

[−ξj(s)]+.(2.18)

We shall call W (t) the reflected process vector of ξ(t) and I(t) the regulator vector of

ξ(t).

Total Workload and Job-Count Process We now proceed with the construction of

the system’s total workload and job-count process vectors, which are the key performance

measures of the system, and the basic ingredients for our limit theorems later on. In this

part we will use the external arrival process from (2.4) as the station arrival process for

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all j ∈ 1, . . . , J. Let’s define Lj(t) and Xj(t) as

Lj(t) ≡ Vj(N(t)) =

N(t)∑i=0

τjvj(i).

Xj(t) = Lj(t)− t.

(2.19)

We shall refer to them as the total workload input process and total workload netflow

process, respectively. From these processes we define Uj(t) and Zj(t), and we shall refer

to them as the total workload process and the total job-count process, respectively.

Uj(t) = Lj(t)−Bj(t) = Xj(t) + Ij(t).

Zj(t) = Nj(t)−Dj(t).(2.20)

When Ij(t) and Dj(t) are the same idle time process and departure process which were

defined for the immediate workload in (2.15) and (2.11)

Ij(t) = t−Bj(t) =∫ t

o1 min

k∈β(j)Qk(s) = 0ds.

Dj(t) = Sj(Bj(t)) = maxk :k∑i=0

τjvj(i) ≤ Bj(t).

(2.21)

The process Uj(t) represents the amount of unfinished work destined for station j that

is present anywhere in the system at time t. In particular, Uj(t) may contain work

corresponding to jobs which at time t are still queued at the station preceding j. The

process Zj(t) represents the total number of jobs in the system at time t that still need

service at station j. It can be seen that Uj(t) is regulated by the same regulator from

Theorem 2.1. In fact it satisfiesUj(t) = Xj(t) + Ij(t) ≥ 0;

dIj(t) ≥ 0, Ij(0) = 0; ∀j

Uj(t)dIj(t) ≥ 0 but Wj(t)dIj(t) = 0.

(2.22)

In other words, it can be seen that Wj(t) is the lower bound of Uj(t) and by (2.16) we

get Uj(t) ≥ Wj(t) ≥ 0 which satisfies the above, dIj(t) ≥ 0; Wj(t)dIj(t) = 0 are satisfied

the same way as in (2.16).

Another interesting relation between the total job-count process and the immediate work-

14

load process can be shown in

Zj(t) = Qk(t) + Zs(k)(t), ∀k ∈ β(j).

⇒ Wj(t) = 0 if mink∈β(j)

Qk(t) = mink∈β(j)

(Zj(t)− Zs(k)(t)) = 0(2.23)

Since it is clear that the total job count for station j equals the sum of the number of

tasks found in buffer k (k ∈ β(j)) and the total job count for the station i (i = s(k)).

From now on throughout this proposal we will denote U(t), X(t), Z(t) as the total workload

vector, total netflow vector and the total job-count vector, respectively.

2.2 State Space Representation

We now define S as the state space of Z(t), i.e. the total job-count vector. Let A be a

K × J matrix whose elements Akj are given by

1, if k ∈ β(j),

−1, if j = s(k),

0, otherwise.

(2.24)

Using the matrix A, we can now define S as the following polyhedral cone in a J-

dimensional space:

S = z ∈ RJ : Az ≥ 0. (2.25)

It can be verified that S is contained in the nonnegative orthant, and that the cone has a

total of K distinct faces, when K is the number of buffers in the system. The state space

k-th face is then defined by

∂Sk = z ∈ RJ : Akz = 0. (2.26)

Where Ak is the k-th row of matrix A.

According to Section 2.1, K can be equal or greater than J . The case in which K > J

is associated with a system containing join nodes, since only join node j can satisfy

|β(j)| > 1. In that case when K > J the state space is contained in a nonsimple

polyhedral cone in the nonnegative orthant.

We will now prove that S satisfies the restriction of the multidimensional regulator and

therefore is acceptable as Z(t)’s state space. We can see from definition (2.24) and (2.23)

15

that Akz (i.e. the k-th row of matrix A multiplied by the vector z) satisfies the following

statements

Akz = zj − zs(k) = Qk when k ∈ β(j),

Akz = Qk ≥ 0 ∀k ∈ 1, . . . , K ⇔ Wj = τj mink∈β(j)

Qk(t) ≥ 0 ∀j ∈ 1, . . . , J,

Ij ↑ if Akz = Qk = 0 for at least one k ∈ β(j).

(2.27)

In other words, the second statement means that the space defined by S = z ∈ RJ :

Az ≥ 0 satisfies the regulator restriction for Wj(t) ≥ 0 ∀j. The third statement shows

that the state space faces ∂Sk = z ∈ RJ : Akz = 0 match the events when dIj > 0 for

some j ∈ 1, . . . , J.We will conclude in the following example for the state space calculation in Fig. 2.2

Figure 2.2: state space example

As we can see, Example 1 shows a fork-join system whose state-space matrix dimen-

sion is 4 × 3, and the associated state space is a nonsimple polyhedral cone in the

R3 space. The cone has a total of 4 distinct faces, which exceeds the space dimension,

meaning that the cone is nonsimple. Example 2 shows a simple tandem system whose

state space is in the R2 space.

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3 Fluid and Diffusion Limits

In this section we will seek heavy traffic limits which involve the convergence of a sequence

of normalized stochastic processes to a limit process.

The traffic intensity at station j is defined as ρj ≡ λτj. The system is said to be stable

if ρj < 1 for all j ∈ 1, . . . , J, and is said to be in heavy traffic if ρj approaches 1 “fast

enough” for at least one j. The precise formulation of our heavy traffic limit theorem

requires the construction of a “sequence of systems”, indexed by n. To construct this

sequence we require sequences of positive constants λ(n), n ≥ 1, τ (n)j , n ≥ 1, j =

1, . . . , J . In the nth system of the sequence, the interarrival time and service times are

taken to be u(n)(i) ≡ u(i)/λ(n) and v(n)j (i) ≡ vj(i)τ

(n)j , respectively. When u(i) : i ≥ 1

and vj(i) : i ≥ 1, j = 1, . . . , J , are sequences of unitized random variables as defined

in Section 2.1, with the restriction that the two sequences are mutually independent

contained with i.i.d random variables with squared coefficients of variation c2a and c2sj,

respectively (the squared coefficients of variation of a random variable is defined to be its

variance divided by the square of its mean).

Section 3.1 begins with the calculation of the fluid limits for the processes of interest

U(t), Z(t),W (t). In Section 3.2 we calculate the diffusion limit for the processes of

interest and prove it is a multidimensional RBM. In Section 3.3 we check the implication

of the diffusion limit on the state space of Z(t) and define the concepts of state-space

collapse and single-bottleneck systems. In Section 3.4 we use the immediate workload

diffusion limit process in order to define the jobs’ throughput time in the system.

3.1 Fluid Limits

Under the definitions of u(i) : i ≥ 1 and vj(i) : i ≥ 1, j = 1, . . . , J , above, it

can be seen that N (n)(t) (2.4) is a renewal process with rate λ(n), and L(n)(t) (2.19) is a

compound renewal process with rate λ(n)τ(n)j = ρ

(n)j . So the fluid limits of the primitives

are

Nn(t) ≡ 1nN(nt)→ λt and Lj

n(t) ≡ 1

nLj(nt)→ ρjt u.o.c, as n→∞ (3.1)

17

Now we will use scaling on the total workload process in the regulator version

Ujn(t) = Xj

n(t) + Ij

n(t);

Xjn(t) = (Lj

n(t)− ρnj t) + (ρnj − 1)t;

Ijn(t) = t− Bj

n(t).

(3.2)

Let n→∞ and we get

Xj(t) = (ρj − 1)t;

Bjn(t)→ (ρj ∧ 1)t ⇒ Ij(t) = (1− ρj)+t = −(ρj − 1)−t;

=⇒ Uj(t) = Xj(t) + Ij(t) = (ρj − 1)+t.

(3.3)

In other words, as seen in (2.22), Wj(t) is the lower bound of Uj(t) and is regulated by

Ij(t). Under the assumption of Qk(0) = 0 ∀k (empty system at t = 0), and ρj ≤ 1 for all

j. The fluid level of Uj(t) and Wj(t) converge together at zero. In conclusion, we get the

following theorem.

Theorem 3.1 Suppose that argument (3.1) hold. Then

(Un, Zn, Bn) → (U , Z, B) u.o.c, as n→∞ (3.4)

if ρj ≤ 1 for all j ∈ 1, . . . , J, then

Bj(t) = ρjt ⇒ Ij(t) = (1− ρj)t;

Xj(t) = (ρj − 1)t;

Uj(t) = (ρj − 1)+t ≡ 0;

Zj(t) = τ−1Uj(t) ≡ 0.

(3.5)

18

3.2 Diffusion Limits

Applying the diffusion scaling on the primitives we get

Nn(t) =√n[Nn(t)− N(t)]; N(t) = λt

Sjn(t) =

√n[Sj

n(t)− Sj(t)]; Sj(t) = τ−1

j t

Vjn(t) =

√n[Vj

n(t)− Vj(t)]; Vj(t) = τjt.

(3.6)

These processes are renewal processes and by applying the functional central limit theorem

for renewal processes, we get the limit processes

N(t) =√λc2a ·BM(0, t)

Sj(t) =√τ−1j c2sj ·BM(0, t)

Vj(t) = τjcsj ·BM(0, t).

(3.7)

Additionally, by applying the scaling on the total workload input process which is a com-

pound renewal process, we get

Lnj (t) =√n [Lnj (t)− Lj(t)]; Lj(t) = ρjt

⇒ Lj(t) = Vj(λt) + τjN(t) ⇒ Lj(t) = Γj ·BM(0, t); Γj = λτ 2j (c2a + c2sj).

(3.8)

Finally, we apply the scaling on the total workload process in the regulator version

Ujn(t) =

√n [Un

j (t)− Uj(t)]; Uj(t) ≡ 0 by fluid limit

⇒ Ujn(t) =

√n [Un

j (t)] =√n [Xj

n(t) + Ij

n(t)];

√nXj

n(t) =

√n (Lj

n(t)− ρnj t) +

√n (ρnj − 1)t = Lj

n(t) +

√n (ρnj − 1)t;

√n Ij

n(t) =

√n [t− Bj

n(t)];

=⇒ Ujn(t) = Lj

n(t) +

√n (ρnj − 1)t+

√n[t− Bj

n(t)].

(3.9)

19

Let n→∞ and we get

Lj(t) = λτ 2j (c2a + c2sj) ·BM(0, t),

θnj =√n (ρnj − 1)→

θj = −∞ if ρj < 1

−∞ < θj ≤ 0 if ρj = 1

√n Ij

n(t) =

√n [t− Bj

n(t)]→ Ij,

(3.10)

When, the first limit is the total workload input diffusion limit calculated in (3.8). The

second limit is called the heavy traffic condition which converges to −∞ < θj ≤ 0 if ρ(n)j

approaches 1 “fast enough” but doesn’t exceed 1. The third limit is the regulator diffusion

limit.

From these limits we can derive the following theorem.

Theorem 3.2 Suppose that argument (3.7) and (3.8) hold. Then,

(Un, Zn) ⇒ (U , Z), as n→∞, (3.11)

where for all j ∈ 1, . . . , J the limit (Uj, Zj) takes the following form (according (3.9)),

depending on the traffic intensity ρj

if ρj < 1 then

√n Xj(t) = BM(−∞,Γj);

Ij(t)→∞;

ddtIj(t) = 1− ρj > 0 ∀t;

=⇒

Uj(t) =

√n Xj(t) +

√n Ij(t) ≡ 0;

Zj(t) = τ−1Uj(t) ≡ 0;

if ρj = 1 then√n Xj(t) = BM(θj,Γj);

=⇒

Uj(t) =

√n Xj(t) +

√n Ij(t) = RBM(θj,Γj);

Zj(t) = τ−1Uj(t) = τ−1 ·RBM(θj,Γj).

(3.12)

20

In other words, the diffusion limit of Uj(t) associated with ρj < 1 converges to RBM with

infinite drift in the direction of zero. Also it can be seen that the dIj(t) > 0 ∀t, which

means that the regulator first-order approximation has a positive rate greater than zero.

This means that Wj(t) ≡ 0 ∀t (by the regulator restriction). As a consequence Uj(t)

converges with its lower bound which is Wj(t) at zero. Contrarily, the diffusion limit of

Uj(t) associated with ρj = 1 converges RBM with drift θj > −∞ and variance Γj.

As a conclusion we get this summary of diffusion limits.

Theorem 3.3 Suppose that arguments (3.7) and (3.8) hold. Then, for all j ∈ j : ρj =

1, the following relation is satisfied

Xj(t) = BM(θj,Γj);

Uj(t) = Xj(t) + Ij(t) = RBM(θj,Γj);

Zj(t) = τ−1Uj(t);

Qk(t) = Zj(t)− Zs(k)(t), Z0(t) ≡ 0, ∀k ∈ β(j);

Qk(t) ≥ 0 ∀t ≥ 0 (by regulator restriction);

Wj(t) = τj mink∈β(j)

Qk(t);

dIj(t) ≥ 0, Ij(0) = 0; ∀j;

Ij ↑ if Akz = Qk = 0 for at least one k ∈ β(j).

(3.13)

In conclusion, from now on let U(t), Z(t), Q(t), W (t) be denoted as the diffusion limits

of the total workload vector, total job-count vector, queue-length vector and immediate

workload vector, respectively.

We will now give a full formulation of the total job-count limit process from which all

the other limits can be derived by the simple transformations in Theorem 3.3. When the

transformations between limit processes is justified by the continuous mapping theorem.

21

Job-Count Diffusion Limit Process-

R = diag(τ−11 , . . . , τ−1

J );

µ = Rθ;

Ω = RΓR;

S = z ∈ RJ : Az ≥ 0;

⇒ Z(t) = multidimensional RBM(S, µ,Ω, R) (3.14)

when for all (i, j)

√n (ρnj − 1)→ θj, Zj ≡ 0 if θj = −∞;

Γij = cov(Zj, Zi) =

0 if Zj ≡ 0 or Zi ≡ 0;

λτjτi(c2a +

cov(Vj ,Vi)

λτjτi) if otherwise;

(3.15)

3.3 State-Space Implications

When using the job-count diffusion limit on real-life systems, it can be seen that the

case when ρj = 1 for all j, is not necessarily a realistic one. In systems with single-class

customers with a single possible route that means µi = µj ∀(i, j) ∈ 1, . . . , J. In real-

life systems we need to apply the diffusion limit on systems in which some of the service

stations are in heavy-traffic and others are not. This case leads to the following theorem

which is a straight consequence of Theorem 3.2.

Theorem 3.4 Suppose that argument (3.7) and (3.8) hold. Let’s define a subset of the

stations’ index Λ which satisfies Λ ⊂ 1, . . . , J : ∀i ∈ Λ ρi = 1, and a subset of the

buffers’ index B which satisfies B ⊂ 1, . . . , K : ∀k ∈ B, ρi = 1 if k ∈ β(i). Then

the diffusion limit of the job-count process is an RBM with the dimension |Λ| whose state

space S = z ∈ RJ : Az ≥ 0 is defined by |Λ| × |B| matrix A. We will refer to the set

of servers indexed by Λ as the system’s critical tasks.

This effect is similar to the effect of non-bottleneck stations collapse shown by Rieman

in [2] for Jackson networks. In fact, it can be shown that the performance measures and

state spaces of the system behave as if the system was a degenerated network containing

22

only servers’ index contained in Λ and buffers’ index contained in B. We shall refer to

this phenomenon as State-Space collapse .

Example 3.1 state space collapse example

In this example we refer to the following system in which Λ = 1, 3, and the second

station’s workload collapses under the diffusion scaling. It can be seen that the original

system degenerates into a simple tandem system containing only the servers in Λ and the

buffers in B (B = 1, 3). Additionally, as the job-count process of station 2 converges

to zero it forces the state space to converge to the Z1, Z3 plane, and reduces system’s

dimension.

23

A special case of the above is the single bottleneck system. Let’s define the diffusion limit

of a job-count process for a single bottleneck system. Assume station i is the bottleneck

station, so the limit process is

R = τ−1i ;

µ = τ−1i θi;

Ω = λ(c2a + c2si);

S = zi ∈ R : zi ≥ 0;

⇒ Z(t) = one-dimensional RBM(S, µ,Ω, R) (3.16)

Which is a one-dimensional RBM whose state space is the nonnegative part of the real

line. As we defined before, the bottleneck station is referred to as the system’s critical

task which determines system performance.

In the following example the fork-join system (a) converges weakly to its bottleneck station

(station 4) which is a G/G/1 system .

Figure 3.1: single bottleneck example

3.4 Heavy Traffic Limit For The Throughput Time

We conclude this section with the definition of throughput time, or sojourn time of a job

in the system; this definition is based on Nguyen in [8].

Let T (t) be the throughput time of the next job to enter the network after time t. A

formal definition will be developed via an inductive definition of intermediate processes

T1(t), . . . , TJ(t), where Tj(t) is interpreted as the throughput time through station j, which

is the time interval between the arrival epoch of a job and when it completes service at

station j.

Let Φ(t) be the random process defined by

Φ(t) ≡N(t)+1∑

1

λ−1u(i). (3.17)

24

One interprets Φ(t) as the arrival epoch of the next job to enter the network after time t.

For each station j ∈ j : d(j) = 1, let

Φj(t) ≡ Φ(t),

Tj(t) ≡ Wj(Φj(t)).

(3.18)

Because station j ∈ j : d(j) = 1 is among the first stations to be visited, Φ(t) is the

arrival time of this job to station j. Furthermore, because jobs are served in a FCFS

manner, the amount of time this job must spend at the station is precisely the amount

of work found at station j immediately after arrival (which includes the service time

associated with the new arrival). Thus Tj(t) is the total sojourn time of the job through

station j.

For other stations in the network, the random processes Φj(t) and Tj(t) are inductively

defined as follows. Suppose that j is a station such that Ti(t) has been defined for each

i ∈ s(β(j)), and set

Φj(t) ≡ Φ(t) + maxi∈s(β(j))

Ti(t),

Tj(t) ≡ maxi∈s(β(j))

Ti(t) +Wj(Φj(t)).

(3.19)

Recall that the arrival time of a job is taken to be the time at which its last component

arrives (If j were a join node, there could be a gap between arrival times of the various

components of the job). Thus, maxi∈s(β(j))

Ti(t) is the amount of time that elapses until the

job “arrives” at station j and Φj(t) is precisely its time of arrival. Hence Tj(t) corresponds

to the throughput time through station j. Setting

T (t) ≡ maxj∈s(β(J+1))

Tj(t), (3.20)

one can conclude that T (t) is the total sojourn time corresponding to the next job to

enter the system after time t.

In conclusion, we derive our last diffusion limit regarding the throughput time process.

Theorem 3.5 Suppose that argument (3.7) and (3.8) hold. Then,

(T (n), T(n)1 , . . . , T

(n)J ) ⇒ (T , T1, . . . , TJ) (3.21)

whereT (t) = max

j∈s(β(J+1))Tj(t);

Tj(t) ≡ maxi∈s(β(j))

Ti(t) + Wj(Φj(t)), T0 ≡ 0,

(3.22)

25

which is the familiar “longest path functional” associated with a critical path analysis of

PERT/CPM.

In the case of a single bottleneck system, it can be seen by using Theorem 3.5 combined

with Theorem 3.4 that the following argument holds:

T (t) = Wj(Φ(t)), (3.23)

which means that the bottleneck station (critical task) alone determines the performance

of the system in reference to the system’s throughput time.

Critical path for bottleneck system

The fork-join system above (system a.) contains three processing routes with the following

sets of servers: 1, 3, 2, 3, 2, 4, respectively. In the limit process the workload in

stations 1, 2, 3 converge to zero, which means, according to (3.23), that the system’s

critical path (or ”longest path”) should converge with probability one to the third route

- servers 2, 4. The bar chart below is an analysis of the system’s simulation runs along

200,000 days which reflect every route factor of time as the critical path for different

arrival rates. It’s clear from the results that as the arrival rate approaches heavy traffic

(ρ→ 1 when λ→ 0.33) the third route becomes the critical path with probability one.

26

4 Brief Discussion on System Control

In this section, we will be looking for methods in which the fluid and diffusion limits

can be used in order to get a better estimation and a better control of real-life systems’

performance measures. This will include a short survey of directions in which work is

being done or may be done by us in this field, as a preparation for our proposed research.

We start in Section 4.1 with a method aimed to dynamically estimate the system through-

put time for arriving customers on the basis of the system occupation on the arrival mo-

ment. This method was introduced by Rieman in [2] and will be referred to as the snap-

shot principle. In Section 4.2 we will discuss the ability to improve the system throughput

time by using priority methods. In Section 4.3 we discuss the ability of improving sys-

tem throughput time by using staffing methods. The method proposed use the notion of

systems’ critical tasks in order to check the workload balancing in the system and offers

appropriate staffing.

4.1 Length of Stay Estimation—The Snapshot Principle

In his paper [2] Reiman stated the following result:

Under suitable conditions, in the limit diffusion time scale the queue length

process does not change during customer’s sojourn in the network.

We refer to this result as the snapshot principle. By using the snapshot principle with

Theorems 3.3 and 3.5 we get the following result, introduced first by Nguyen in [8]. For

a customer arriving at Φ(t)

Q(Φ(t)) is given by a deterministic vector Q

Wj(Φ(t)) = Wj = τj mink∈β(j)

Qk, determinstic for all j;

⇒

T (Φ(t)) = max

j∈s(β(J+1))Tj(Φ(t));

Tj(Φ(t)) ≡ maxi∈s(β(j))

Ti(Φ(t)) +Wj.

(4.1)

In other words, if one takes a “snapshot” of the system at the time of the job’s arrival,

Φ(t), the deterministic queue length vector observed in this “snapshot” can be used to

27

estimate a deterministic counterpart for the immediate workload vector experienced by

the customer throughout the processing in the system. Therefore, the throughput time can

be calculated by Theorem 3.5 as if W is a deterministic vector. This result reduces fork-

join throughput time analysis to a sample-path by sample-path analysis of a PERT/CPM

network, in which workload levels take the place of task time.

This method of estimation was tested in matlab simulation runs for different networks

and was found efficient in the sense of small RMSE.

Snapshot Example

Figure 4.1: snapshot simulation results

This is an analysis of simulation runs in which the snapshot method was checked on the

system in Fig. 2.1. These graphs compare the customer’s actual LOS in graph a; the

offline calculation of each customer’s LOS according to (4.1) in graph b; and the estimator

error distribution in graph c.

28

4.2 Priorities Control

In his paper [1] Cohen examined the critical-chain (CC) method and other control mecha-

nisms in a multi-project environment. His research was focused on priority discipline and

buffer management control methods. Included in his research are methods without buffer

overflow such as critical-chain and minimum slack, and methods with buffer overflow (i.e.

when the controlled buffer is full the arriving customers are forced to leave without being

treated) such as, a constant number of projects in process and queue size control. The

conclusion of the research was that there are differences in system performance between

the various methods in heavy traffic, which means that control can possibly improve sys-

tem performance.

In our proposal we will not allow customers to leave the system without being treated,

but it can be simply seen that these methods of buffer overflow do “improve” system

performance in heavy traffic by not letting the system approach heavy traffic. In these

methods the control keeps ρj < 1 ∀j (by definition) and therefore keeps queue length

process and throughput time well behaved and hence converges to zero under diffusion

scaling. The disadvantage of these methods is that the fraction of clients leaving the

system without being treated will converge to one in heavy traffic.

We would like to focus now on methods which don’t allow customers to leave the system

without being treated. We claim that in the heavy traffic limit of single-type customers

the optimal priority discipline is FCFS and other priority disciplines can only be equal to

or reduce system performance, in contradiction to Cohen’s paper. This claim is based on

the sequence of the following arguments, which will be given here without proof.

1. Under suitable conditions and according to Section 4.1 we can argue the following: In

each time point of the system’s operation, the group of customers who are currently

being served will experience a deterministic immediate workload vector, which means

that there will be one deterministic critical path that will determine the customers

LOS in the system. According to this critical path the LOS will behave as a pure

sequential system. Moreover, in the case of a single bottleneck system according to

(3.23) the system LOS will behave as a simple G/G/1 system.

2. In a sequential system of G/G/1 stations with homogeneous customers the LOS

time is invariant to the priority discipline. Let’s assume that we want to rearrange

a group of N customers in order to get minimum average LOS time. Let us define

customer index i ∈ 1, . . . , N in the same order as the client processing order. So

29

for any G/G/1 station

T =1

N

N∑i=1

i∑j=1

τj, (4.2)

when τj is customer j mean service time, and for a sequence of G/G/1 we get

T = T1 + T2 + . . .. For homogeneous customers τi = τj ∀(i, j) ∈ 1, . . . , N.That means that T is invariant to the client processing order i, and that in the

deterministic critical path case priority discipline doesn’t change the average LOS

time.

3. Another important factor to be considered is the system’s synchronization require-

ments for the join nodes. Components arriving to a join node are joined only if

they correspond to the same job. As a consequence, the waiting time of a job since

its first component arrives and until it may start processing at join node j, can be

defined in υj = maxi∈β(j)

Ti − mini∈β(j)

Ti. It can be seen that υj depends on the synchro-

nization of a customer’s processing order in different paths through the system, and

on the variance of the priority discipline. In that sense FCFS has the advantage of

low variance discipline and that it keeps the order of the customer’s arrival. For an

opposite case, one can think of unsynchronized preemptive priority (in every path

different client types get the lower priority) whose variance in heavy traffic tends to

go to infinity, and it can be seen that the resulting waiting time in the joins nodes

also goes to infinity, which means that the join node server is idle in times when the

buffers are full (with components of different jobs).

This claim was verified in matlab simulation runs for different networks and different

priority disciplines.

4.3 Staffing-Control

When taking into consideration the conclusions from the total workload diffusion limit in

Theorem 3.2 and especially the state-space collapse notion defined in Theorem 3.4, we can

look from different points of view on the concept of a system’s workload balancing. When

the set of “critical tasks” Λ (see Section 3.3) satisfies |Λ| < J (when J represents the

actual number of service stations in the system), or in a more extreme case, if |Λ| = 1, a

single bottleneck system, then we can argue that the staffing is not efficiently distributed

according to the expected workload. In this case we can say that “most of the work is

assigned to only a few stations”.

We propose a staffing method which inforce the following rule - ρj approach 1 uniformly

30

fast for all j. In systems in which all the customers have the same single possible precedence

constraints scheme (single deterministic Pij matrix, (2.1)). Then the arrival rate is uniform

in all of the stations’ entrances, which leads to the following staffing rule

Route Balancing staffing Given a system containing N servers and J separate tasks

(J stations) which need to be carried out upon each arriving customer (naturally N ≥ J),

and given that the network has an input stream of jobs with a single possible precedence

constraints scheme. The staffing Nj should be the result of the following IP problem-

Let us define λjcritical =Nj

τj∀j ∈ 1, . . . , J and λcritical = 1

J

J∑j=1

λjcritical, then

minNj

√√√√ 1

J − 1

J∑j=1

(λjcritical − λcritical)2

s.t

J∑j=1

Nj = N.

Nj ≥ 0 integer, ∀j ∈ 1, . . . , J.

(4.3)

It is easy to see that this method is optimal in the sense that it sets the heavy traffic

condition (ρj = 1 for at least one j) of the system to the highest arrival rate achievable

for all staffing methods. With one more advantage, this method may be applied off-line

in the design of the system independently from the unknown arrival rate λ, and doesn’t

need to change on-line unless the servers’ service rates vary with time.

This method of staffing was checked in matlab simulation runs and has shown improve-

ment in system performance in heavy and light traffic. For example, we ran this method

on the system in Fig. 2.1 with the server properties (τj, Nj) = (6, 3), (5, 2), (4, 3), (3, 1),respectively. It can be checked that those properties make the system unbalanced.

After applying the route balancing we get the following server properties: (τj, Nj) =

(6, 3), (5, 2), (4, 2), (3, 2), which means we only moved one server from station 3 to sta-

tion 4.

The effect of this change can be seen in the following graph (Fig. 4.2) of the system’s

throughput time vs. arrival rate.

31

Figure 4.2: route-balancing simulation example

In other words, the arrival rate in which the system enter heavy traffic have changed

from λ = 0.33 to λ = 0.4, but the improvement in the system’s throughput time is not

restricted only to the heavy traffic region, since improvement can be seen in light traffic

region as λ = 0.28 and less.

32

5 Proposed Research

In this section we propose directions that expand the work described above. These di-

rections are straightforward consequence of the previous discussion on analyzing system

performance measures using fluid and diffusion limits, and applying control methods which

are based on the limiting processes. The following proposals can and will be applied to

real-life processes associated with hospital performance measures and management poli-

cies.

5.1 Performance Analysis

• Single-Type and Single-Bottleneck Systems As described in Section 3.3, we

claim that the limit process collapses into a single G/G/1 station. In our work

we shall start by verifying that claim, and also define the notions of critical task

and critical path and their implications on system performance. We shall research

further the effects of synchronization gaps on system performance as discussed in

Section 4.2, and the dependence between synchronization requirements and the limit

process after system collapse.

• Multi-Type and Multi-Route Customers In her subsequent work (see [9]),

Nguyen extended system representation in order to include feedforward networks

of single stations with FCFS discipline populated by multiple job types. In her

paper it was shown that the resulting polyhedral region has many more faces than

its homogeneous counterpart and that the description of the state space becomes

vastly more complicated in this setting. It can be seen that the rise in system’s

complexity is a direct consequence of the system’s synchronization requirements,

which were introduced in Section 4.2.

In Reiman [2] and Peterson [10], it was shown that in Jackson networks exists state

space collapse with respect to customer types. In their work, they show that the

high priority customer’s workload process converges to zero under diffusion scaling.

In our further work we propose to check the application of that result in fork-

join networks and reduce system’s complexity by using a synchronized preemptive

priority discipline. In this method, theoretically we can degenerate heterogeneous

single-bottleneck systems into homogeneous single-type single-station systems in the

same manner as in the state space collapse introduced in Section 3.3.

• Fork-Join and Jackson Mixture Additional extension of the model is to relax

the routing discipline into allowing probability-based routing (as in Jackson net-

33

works) combined with deterministic routing (as in Fork-Join). In order to do that,

we will have to redefine our system and state-space representation. Additionally,

we will have to define the conditions for heavy traffic and stability, and redevelop

the system’s performance-measures limit processes.

• Include Feedback in Routing Disciplines Additional extension of the model

can be to relax the routing discipline into including feedback routing, meaning that

the same task can be repeated several times. Recent results [4] show that these

networks can be unstable in the sense that the total number of jobs in the network

explodes as time goes to infinity even if the traffic intensity at each station is less

than 1. If time allows, we will further research and redefine conditions for heavy

traffic and stability in this class of systems.

The above model relaxations are very important in order to accommodate and analyze

complicated systems such as in the customer flows in hospitals, where multi-type cus-

tomers and complex routing are prevalent.

5.2 Stochastic Control

• Priority Control in Single-Type Customer Systems As we discussed in Sec-

tion 4.2, there is a lot of work left to be done in order to understand the effect of

priority disciplines on system performance, and the relation between the priority

discipline and system synchronization requirements. All this is necessary in order

to develop optimal system priority control.

• Priority Control in Multi-Type Customer Systems As discussed within the

performance analysis section, using priority and especially preemptive discipline can

possibly reduce system’s complexity and treat synchronization gaps. However, the

question remains can we achieve improvement in system performance? We propose

to examine static or dynamic partition scheme of preemptive-priority classes, which

is designed to achieve preferable performance. For example, let us define a deter-

ministic cycle, in which every customer type gets a predefined factor of time as

the lowest preemptive priority. This is a version of “Round-Robin” for the priority

discipline. It can be verified that the customers’ priority duty-cycle determines each

customer fraction of the total queue length at the heavy traffic limit system.

• Staffing-Control We propose to verify and extend the staffing methods which are

based on the limit processes analysis such as the Route Balancing Staffing proposed

34

in Section 4.3. We can also examine dynamic closed-loop staffing based on the

snapshot principle.

• Routing Control with Diversely Skilled Servers In many real-life systems,

servers can be used in more flexible disciplines in which the same server can do

different tasks with different service times associated with each task. Building an

appropriate model to such systems, one can think of all the servers as a pool of

diversely skilled servers and of each task as a buffer of work waiting to be treated.

In this model, tasks are routed from the buffers to free servers in the pool by

some routing discipline. In this framework, one can ask what the optimal routing

discipline is in order to minimize queue lengths or throughput times.

• Asymptotically Optimal Control As shown in Section 3.2, the diffusion scaling

can be interpreted as1

nUnj (nt) = Uj(t) +

1√nUj(t). (5.1)

In other words, the fluid limit and the diffusion limit can be interpreted as the first-

and second-order approximation of the process of interest, respectively. Keeping

that in mind, we can develop a control model based on the appropriate process

of interest in which we control the system through staffing / routing / preemptive

classes partition. The goal is to construct a model, which can be used in order to

achieve optimality in the sense of first and second order approximations.

The above questions are very important in hospitals. Indeed, typically customers’ flow

into the system varies over time in rates and proportions, and flexibility in staffing is

achieved by several methods such as- “floater” nurses that move from one ward to the

other, or a nurses’ pool which is used in high occupation situations.

35

References

[1] Cohen I., Mandelbaum A., Shtub A., Multi-project scheduling and control: a process-

based comparative study of the critical chain methodology and some alternatives,

Project Management Journal 35 (2004), 39–50. 1.2, 4.2

[2] Reiman M., Simon B., A network of proiority queues in heavy traffic: One bottleneck

station, Queueing Systems 6 (1990), 33–58. 1.2, 3.3, 4, 4.1, 5.1

[3] Chen H., Yao D., Fundamentals of queueing networks, Springer-Vrelag New York,Inc

(2001). 1.2

[4] Banks J., Dai J. G., Simulation studies of multiclass queueing networks, IIE Trans-

actions 29 (1997), 213–219. 1.2, 5.1

[5] Harrison J., Brownian motion and stochstic flow systems, Wiley Series in Probability

and Mathematical Statistics (1985). 1.2

[6] , A broader view of brownian networks, Stanford University (2002). 1.2

[7] Larson R., Cahn M., Shell M., Improving the N.Y.C A-to-A system, Interfaces 23

(1993), 76–96. 1.1

[8] Nguyen V., Processing networks with parallel and sequential tasks: Heavy traffic anal-

ysis and brownian limits, The Annals of Applied Probability 3 (1993), 28–55. 1.2,

3.4, 4.1

[9] , The troubles with diversity: Fork-join networks with heterogeneous customer

population, The Annals of Applied Probability 4 (1994), 1–25. 1.2, 5.1

[10] Peterson W., Diffusion approximation for networks of queues with multiple customer

types, PhD Thesis, Stanford University (1985). 1.2, 5.1

[11] Whitt W., Weak convergence theorems for priority queues: Preemptive-resume dis-

cipline, Journal of Appliad Probability 8 (1971), 74–94. 1.2

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