Pedestrian Performance Measures of an M/G/C/C State Dependent Queuing Network in Emergency
Occupant safety is a major concern for a building in case
of emergencies such as fire, bomb blast or earthquake. In such emergency situations,
a quick and jam-free evacuation of pedestrian trapped in a network of corridors
in buildings is most important. In this paper, we compared the throughputs for
two different restrictions on the flow direction with the unrestricted flow
for a facility. In this facility, occupants need to go through some source corridors
which consists of multiple sources and from source corridors they can choose
their nearest exiting corridors. The result indicates that in emergency situation
a restricted flow direction of pedestrian increases the throughput of the network
which eventually decreases the average evacuation time.
Received: November 30, 2012;
Accepted: March 13, 2013;
Published: April 22, 2013
Congestion in a pedestrian facility occurs due to increased density of occupants
which decays the service rate. Congestion in pedestrian traffic networks (Mitchell
and Smith, 2001; Yuhaski and Smith, 1989), vehicular
traffic networks (Jain and Smith, 1997), in material
handling systems (Bedell and Smith, 2012; Smith
and Kerbache, 2011) and many other systems where service rate decreases
with increased density of occupants, can be appropriately modelled by using
state dependent queuing networks.
Cruz et al. (2005a,b), Cruz
and Smith (2007), Mitchell and Smith (2001), Smith
(1991), Yuhaski and Smith (1989) and others had
used state dependent queuing models to capture the congestion in pedestrian
traffic flow with arbitrary topologies and for different types of multi-storied
buildings. Kawsar et al. (2012) also used M/G/C/C
state dependent queueing models to evaluate the performance measures of a complex
facility and showed that in case of an emergency throughput can be increased
by restricting the flow direction and controlling the arrival rate to each of
its source corridors.
In this study, we considered the same facility as Kawsar
et al. (2012) where occupants through the source corridors come from
multiple sources and egress through the nearest corridor. The objective of this
study is to estimate the different performance measures along with the average
evacuation time for unrestricted and restricted flows during egress from the
facility. These measures can be used to evaluate the optimal set-up for which
the maximum throughput can be obtained.
MATERIALS AND METHODS
Analytical pedestrian flow model: Based on some empirical studies, Tregenza
(1976) presents a number of relationships between the walking speed of a
pedestrian and the crowd density which is re-created in Fig. 1.
According to Fruin (1971) uni-directional flow models
can be used to capture the bi-directional and multi-directional flows of occupants
during an evacuation and the flow relationships are similar for stairwells and
plane movements. Linear and exponential models for uni-directional walking speed
has been developed by Yuhaski and Smith (1989) as follows:
||Shape and scale parameters for the exponential
||Average walking speed for n occupants in a corridor
||Average walking speed when crowd density is 2 ped m-2
= 0.64 m sec-1
||Average walking speed when crowd density is 4 ped m-2
= 0.25 m sec-1
||V1 = Average walking speed of a lone occupant = 1.5 m sec-1
||Number of occupants in a corridor
||Length of the corridor
||Width of the corridor
Cheah (1990) provided exponential walking speed models
for bi- and multi-directional corridor which are similar in the form to the
uni-directional model. In this study, the analysis is carried out solely for
uni-directional traffic flows through corridors using the exponential pedestrian
M/G/C/C state dependent queuing system: In M/G/C/C state dependent queueing
model the arrival rate of pedestrians are assumed to be Markovian. The total
number of pedestrians that can enter the system is equal to the capacity of
the corridor which is also the number of servers, since the corridor behaves
as a server to the occupants. The service rate is the rate at which pedestrians
pass across the entire length of the corridor. This rate is dependent on the
number of occupants (n) within the corridor and follows a general distribution
G. Hence, the queuing model is state dependent. Cheah (1990)
showed that M/M/C/C and M/G/C/C state dependent queues are stochastically equivalent
and developed the limiting probabilities for the number of pedestrians in an
M/G/C/C state dependent queuing model as follows:
In this model, E(S) is the expected service time of a lone occupant in a corridor of length L that is:
Pn is the probability when there are n occupants in the corridor and P0 is the probability of the corridor being empty. The service rate, f(n), is the ratio of the average walking speed of n pedestrians (Vn) in the corridor to that of a lone pedestrian (V1) that is:
When a pedestrian attempts to enter a corridor, but cannot because the corridor is currently at capacity then the blocking occurs and the probability of such blocking is equal to Pn where n equals C, the capacity of the corridor. The different performance measures can be computed as:
||Steady state throughput through corridor
||Expected number of occupants in the system
||Expected service time in seconds
Following the work of Yuhaski and Smith (1989), Kawsar
et al. (2012) modeled corridor with multiple arrival sources and
represented the facility as a network of series, split, merge or a combination
of these topologies.
Description of the facility: In this study, the same facility as Kawsar
et al. (2012) is considered which is presented in Fig.
2. The numbers represent the corridors, the alphabets S, T, U, V, W, X,
Y and Z represent the different seating arrangements and A, B, C
and D are the exits to other corridors. The number of seats in S, T, U,
V, W, X, Y and Z are 220, 220, 309, 259, 100, 55, 98 and 77, respectively. The
main exit door situated at the end of corridor 3 is modelled as one large door.
Smith (2009) showed that node splitting at double width
exits allowing for a support beam has no effect on throughput. Each of corridors
10 and 11 is modelled as a single corridor with 20 and 16 arrival sources, respectively
and each of corridors 6, 7, 8 and 9 as a single corridor with three arrival
sources. In cases when there is a split, the throughput from the splitting corridor
is divided according to the probabilities of the branches. The arrival rate
to a merging corridor equals the sum of the throughputs of the previous corridors.
Table 1 presents the dimension, number of sources and average travelling distance of the source corridors. Considering the arrival rate of a source corridor as the sum of the arrival rates of all related sources and the average travelling distance, the different performance measures of these corridors can be calculated. The different dimensions of the exiting corridors are presented in Table 2.
||Dimensions, number of sources and average travelling distance
of source corridors
|| Simplified representation of corridors of the facility
|| Dimensions of the exiting corridors
RESULTS AND DISCUSSION
The corridor capacity depends on its size and the throughput of a corridor
depends on the arrival rate. Considering the whole facility as a network of
corridors, the throughputs of the source and exiting corridors for different
arrival rates are presented in Fig. 3 and 4,
respectively. The patterns of these graphs are similar with that obtained by
Mitchell and Smith (2001). However, the magnitudes
of the curves vary as the throughput depends on the capacity of corridor as
well. From Fig. 3, we observe that the highest throughput
for corridors 6 and 7 occur for an arrival rate of more than 14 ped sec-1.
Similarly for corridors 8 and 9, this rate is more than 10 ped sec-1
and for corridors 10 and 11 it is more than 6 ped sec-1. From Fig.
4 it is observed that the throughputs for some of the exiting corridors
coincide, because of approximately same size of the corridors. These are corridors
1 and 5; corridors 2 and 4; corridors 12 and 13 and corridor 14 and 15.
To increase the throughput of the entire system we have put a new restriction
in place along with the restriction used previously by Kawsar
et al. (2012). We restrict occupants from corridor 10 to exit only
through corridors 12 and 13 and occupants from corridor 11 to exit only through
corridors 14 and 15. As Xiang (2007) observed that people
looked for the nearest exit which is shorter, occupants from all other source
corridors are assumed to choose their nearest corridor to exit. Under this restriction,
the average travelling distance for the occupants in corridor 10 and 11 is changed
to 4.95 and 4.095 m, respectively. Also the arrival rates for corridors 12,
13, 14 and 15 are changed. The new arrival rate for each of corridors 12 and
13 is now half of the throughput of corridor 10 and that for each of corridors
14 and 15 is half of the throughput of corridor 11. Also the new arrival rate
for corridor 3 is half of the throughput of corridor 7 plus half of the throughput
of corridor 8. The throughputs for the source and exiting corridors under the
restrictions are presented in Fig. 5 and 6,
respectively. It shows that the new restriction decreases the throughputs of
both source corridors 10 and 11. This happens because the restriction increases
average travelling distance and arrival rate.
||Throughputs of source corridors for varying arrival rates
||Throughputs of exiting corridors for varying arrival rates
||Throughputs of source corridors for varying arrival rates
||Throughputs of exiting corridors for varying arrival rates
||Expected number of occupants of the source corridors for unrestricted
This coincides with the findings of Lo and Fang (2000)
and Cruz et al. (2005a,b).
Lo and Fang (2000) showed travel distance decreases
flow rate at the congested points and Cruz et al. (2005a,b)
showed that throughput and expected number of customers are closely dependent
upon the arrival rate. The decrease in the throughputs of the source corridors
results a low occupant density in the exiting corridors and decreases the evacuation
time. Liu et al. (2009) also found a similar
result that occupant density around the exit influences the evacuation time.
The expected number of occupants and the expected service time for each of
the source corridors against varying arrival rate for the unrestricted and restricted
flows are presented in Fig. 7-10. The figure
shows that for unrestricted flow, both corridors 10 and 11 reaches the capacity
for an arrival rate slightly greater than 8 ped sec-1 and the expected
service times are approximately 17 and 14 sec, respectively. Under the restriction,
for an arrival rate of approximately 4.5 ped sec-1 both corridors
reach the capacity and the expected service time increased to approximately
31 and 26 sec, respectively.
||Expected number of occupants of the source corridors under
||Expected service time for the source corridors for unrestricted
||Expected service time for the source corridors under restricted
This reduces congestion in exiting corridors and results an increase in the
throughputs of the exiting corridors. A similar result was obtained by Cruz
(2009) for a finite single server general queuing network.
||Total throughput of the facility for unrestricted and restricted
The overall throughput of the system for the restricted along with the unrestricted flows is presented in Fig. 11. It is observed that for the unrestricted flow, the curve has two peaks. The throughput reaches its maximum value 13.86 ped sec-1 when the arrival rate is approximately 2.60 ped sec-1. The second peak occurs at 13.26 ped sec-1 for arrival rate of 6.30 ped sec-1. As the arrival rate increases from 6.30 ped sec-1, the congestion begins to increase and the service rate of each occupants decreases. Thus, despite an increased arrival rate within the corridors, the decreased service rate of each occupant affects the overall throughput as arrival rate increases past 6.30 ped sec-1. As the number of occupants in corridors approaches to corridor capacity, an increase in arrival rate does not have a significant effect on the overall throughput and the throughput appears to reach a limit as arrival rate increases.
For restricted flow, the throughput reaches the maximum value of 15.35 ped
sec-1 when arrival rate is approximately 2.60 ped sec-1.
As the arrival rate increases from 2.60 ped sec-1 the throughput
decreases and for an arrival rate of 3.90 ped sec-1 it reaches the
limit which is slightly greater than 13 ped sec-1. That is, if the
arrival rate increases past 3.90 ped sec-1, the throughput remains
approximately same. Such findings agree with the findings of Mitchell
and Smith (2001). The figure shows that the overall throughput increases
for the restricted flows compared to the unrestricted flow.
Figure 12 shows the expected evacuation time for the different types of flows for varying arrival rates. It shows that the expected evacuation time is less for the restricted flow compared to that for unrestricted flow. Under restricted flow, the highest expected evacuation time is achieved for an arrival rate of approximately 4 ped sec-1 and it approaches to the limit as arrival rate increases past 4 ped sec-1.
||Expected evacuation time from the facility for unrestricted
and restricted flows
It can be observed that restrictions on the flow direction and arrival rates
of the source corridors reduce the average evacuation time. If from all the
seating arrangements occupants rush to exit, then there will be a high probability
of congestion. Since majority of the occupants sit surrounding corridors 10
and 11 and they need to travel a long way for the exit, the arrival rate to
these corridors need to be controlled in such a way that the highest throughputs
from the exiting corridors can be maintained. Shende et
al. (2011) has also achieved smooth flow of pedestrians using control-algorithm.
In case of an emergency, occupants from seating arrangements S and T may be
suggested to go through corridors 14 and 15 and from arrangements U and V they
may be suggested to go through corridors 12 and 13. While it may seem that it
will decrease throughputs from source corridors, there will be less congestion
in exiting corridors and will reduce the average evacuation time. Xiang
(2007) also showed the similar result for lecture theater type rooms that
when there are congestions in the aisle area connecting with each row, the doorway
was not fully occupied. In addition, Shende (2008) showed
that for uncontrolled case, flow got interrupted within a few seconds of starting,
however it did not happen in the controlled case and revealed the usefulness
of the pedestrian control approach.
We have used state dependent M/G/C/C queuing model to capture the bottleneck effects of pedestrian flow within the corridors of a facility involving combinations of merges and splits. The different performance measures of all the corridors of this hall room during egress have been computed. These measures are used to compute the evacuation time of the hall room in case of an emergency and can be used to evaluate the optimal internal set up for which the maximum throughput can be obtained. We have calculated the different performance measures for restricted and unrestricted flows. The higher arrival rate to the source corridor is found as a cause of higher congestion in the exiting corridors. The congestion may be controlled by putting some restrictions on arrival to the source corridors and on travelling direction from source corridor to exiting corridors. Further extensions of this work will be to investigate the optimal internal set up for such facilities.
This study was supported by the Research University (RU) Grant Scheme, [Acct. No.: 1001/PJJAUH/ 811097], Universiti Sains Malaysia. L.A. Kawsar wishes to thank Universiti Sains Malaysia for the financial support (USM Fellowship).
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