A design to minimize congestion around the Kaʹaba

One of the rituals of the Moslem pilgrimage to Makkah is Tawaf. For Tawaf, a pilgrim walks seven times around the Kaʹaba. The Kaʹaba is a cubic structure. The area available around the Kaʹaba is relatively small and is usually congested. People arrive at random to perform this ritual and immediately join the crowd, further congesting the area. The finishing pilgrims jostle their way out causing traffic conflict. This situation causes safety and religious problems in addition to delay. This study suggests directing walking pilgrims along a path spiralling seven times around the Kaʹaba. A model is developed to minimize the total waiting time in the system as a function of width of the spiral path and the level of service (that is, flow rate, speed, space per person). An example is presented using historical data. Furthermore, we obtain the optimal width of spiral as the population increases. We also consider the system where a separate lane is added for movement of wheel chairs.